Wexon Handbook for selection of motors Dunkermotoren 2020 by Buorre

Handbook for selection of motors

Editorial Dunkermotoren GmbH is an internationally operating company and a leader in the field of lowpower electrical motors. Dunkermotoren GmbH offers a wide range of gearboxes and electrical motors based on a sophisticated modular system. In combination with many years of experience in development, manufacturing and design of electrical motors Dunkermotoren GmbH presents itself as a supplier of simple electric motors as well as a partner for the realization of complex motor technologies. The current formulary was created by specialists to support users of electrical motors. It helps to understand electrical motors in their application and assists in the selection of motor components. The systematic set-up of the formulary facilitates the introduction into the subject matter and makes this formulary a valuable tool for apprentices, students, career beginners and experienced professionals. The breadth of motor components and solutions considered satisfies the requirements of engineers who have to design, select or adapt their motors in their daily work. For practitioners, the collection of examples serves as a guide for their own calculations, thus facilitating the application of the formulas for real design tasks. Electrical motors are continually evolving, and so the present formulary is subject to constant revision. If you miss any content or have suggestions and wishes for future editions of the formulary, we are looking forward to your feedback.

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Content

Content 1 Systematic ��6 2

Selection of the optimum motor technology ��7

2.1

Overview of possible motor technologies. ��7

2.1.1

Classification by application ��7

2.1.2

Classification by operational quadrant ��8

2.1.3

Classification according to technical realisation ��8

2.2

Selection of the optimum motor technology ��9

2.3

Electrical motors from Dunkermotoren ��14

2.3.1

Permanent Magnet DC Motors - Series GR ��14

2.3.2

Brushless DC Motors - Series BG ��14

2.3.3

Asynchronous AC Motors - Series KD/ DR ��15

2.3.4

Linear Systems - Series ST | CASM | LS. ��15

2.4

Example: Pump motor ��15

2.5

Example: Door motor ��16

Description of the load cycle ��22

3.1

Kinematic equations. ��22

3.2

Motion laws. ��25

3.2.1

Motion laws for constant speed motors ��23

3.2.2

Motion laws for variable speed motors ��23

3.2.3

Motion laws for servo motors ��24

3.3

Masses and moments of inertia. ��32

3.4

Load forces and load torques ��34

3.5

Example: Pump motor ��36

3.6

Example: Door motor. ��36

Motion conversion with gears. ��37

4.1

Rotating axes ��38

4.1.1

Rotating load ��38

4.1.2 Pump ��39 4.1.3 Fan. ��40 4.2

Linear axes ��41

4.2.1 Spindle ��41 4.2.2

Conveyor belt. ��42

Content

4.2.3

Wire rope and belt. ��43

4.2.4

Toothed rack. ��44

4.2.5 Hoist ��44 4.2.6

Running gear ��45

4.3

Rotating gear ��46

4.3.1

Mathematical description and parameters ��46

4.3.2

Selection of the optimum gear ratio ��47

4.3.3

Selection of the optimum gearbox type ��58

4.3.4

Selection of the gearbox ��54

4.4

Example: Pump motor ��57

4.5

Example: Door motor ��58

Design of the motor ��67

5.1

Characteristics of electrical motors ��67

5.2

Constant speed motors ��68

5.2.1

Constant speed motors with permanent magnet DC motor ��69

5.2.2

Constant speed motors with asynchronous motor ��71

5.3

Variable speed motors and Servo motors ��72

5.3.1

Selection of the motor according to mechanical parameters ��72

5.3.2

Selection of the motor according to thermal characteristics ��74

5.3.3

Selection of the encoder ��76

5.3.4

Encoder and brakes from Dunkermotoren ��77

5.4

Example: Pump motor ��77

5.5

Example: Door motor ��78

Selection of the control unit ��81

6.1

Characteristics and selection of the control unit ��81

6.2

Integrated and external control units from Dunkermotoren ��83

6.2.1

Controllers - Series BGE/ DME ��83

6.2.2

Motion control 4.0 ��84

6.3

Example: Door motor ��84

List of figures. ��86

List of tables ��86

Systematic

1 Systematic Electrical motors consist of a variety of components which are to be treated mathematically. •

The gearbox is a mechanical converter. It adjusts the physical quantities, such as the

•

The electrical motor serves as an energy converter which converts the electrical energy

1 Systematic speed drives and theconsist torque, of delivered by the motor to the requirements the working machine. Electrical a variety of components which are toofbe treated mathematically.

• Theinto gearbox is a mechanical converter. It adjusts the physical quantities, such as th supplied mechanical energy. •

and the torque, delivered the motor to the requirements of theposition working machine. The motor encoder detects actualby motion variables such as speed, velocity, and

The electrical motor serves as an energy converter which converts the electrical en supplied into mechanical energy. The brake prevents movement of the motor when the control unit is switched off. • The motor encoder detects actual motion variables such as speed, velocity, position The control is an electric energy and controls the motion variables of the makesunit them available to the converter signal electronics. electrical • The motor. brake prevents movement of the motor when the control unit is switched off. • The control unit is an electric energy converter and controls the motion variables of electrical motor. •

makes them available to the signal electronics. • •

Switching and protection devices, if necessary transformer

Figure 1 Components of an electrical drive The sequence inofwhich the individual Figure 1 Components an electrical motor

components are considered is based on the procedure design of electrical drives. It is carried out in steps which build on each other and, starting fro working machine, passes through all components of the electrical drive.

The sequence in which the individual components are considered is based on the procedure for the design of electrical motors. It is carried out in steps which build on each other and, starting

1. Selection the optimum drive technology from the working machine, passes through allof components of the electrical motor. 2. Definition of the motion sequence and the load 6

3. Selection of the gear 4. Selection of the motor and the encoder

Systematic

1. Selection of the optimum motor technology 2. Definition of the motion sequence and the load 3. Selection of the gear 4. Selection of the motor and the encoder 5. Selection of the control unit 6. Check of the costs and, if necessary, repeating steps 1 to 6 with another motor technology Figure 2 Procedure for the design of electrical motors

Selection of the optimum motor technology

2.1 2.1.1

Overview of possible motor technologies Classification by application

For the design of electrical motors, it is helpful to classify them into three categories with regard to the requirements for speed adjustability. Constant speed motors

Variable speed motors

Servo motors

Constant speed motors are

Variable speed motors are

Servo motors are designed

operated at a fixed speed.

adjustable in their speed.

to perform positioning tasks.

They only have facilities for

In addition to the electrical

They have a high acceleration

switching on and off, as well

motor, they have a control

capability and a high overload

as for protection against

unit that is responsible for

capacity. In addition to the

overload. There is no speed

the speed adjustment.

electrical motor (servomotor),

adjustment device.

they have a control unit and position sensor (closed loop).

Typical applications

» Pumps

» Trolleys

» Door motors

» Fans

» Hoists

» Motors in packaging and

» Conveyors

filling machines

» Pull motors and winders

» Feed motors in machine

for continuous webs (film,

tools

paper, textile fibres, wire)

» Robot motors

Table 1 Classification of electrical motors with regard to speed adjustability

machines • Conveyors • Feed drives in machine tools • Pull drives and winders for continuous webs (film, paper, • Robot drives textile fibres,motor wire) technology Selection of the optimum able 1 Classification of electrical drives with regard to speed adjustability Conveyors

2.1.2

Classification by operational quadrant

.1.2 Classification by operational quadrant Electrical drives have the ability of reversing the direction of rotation and the ability of energy recovery. hese characteristics of an electrical drive are shown in a speed-torque diagram. Depending on the ign of the speed and the torque, 4 operating quadrants result. of reversing the direction of Electrical motors have the ability • •

rotation and the ability of energy

recovery. These of drive an electrical motor In motor mode quadrants, thecharacteristics speed and torque of the have the same sign. are shown In generator mode quadrants, the speed and the torque are directed in opposite directions.

in a speed-torque diagram.

Depending on the sign of the speed and the torque, four operating quadrants result.

As a positive direction of rotation, the clockwise rotation is defined with respect to the motor shaft.

In motor mode quadrants, the speed and torque of the motor have the same direction. In generator mode quadrants, the speed and the torque are directed in opposite directions. As a positive direction of rotation, the clockwise rotation is defined with respect to the motor shaft. Depending on the design of the control unit, electrical motors operate only in the first quadrant (e.g. pumps) or in all four quadrants

igure 3 Operational quadrants of electrical drives

(e.g. hoists).

Figure 3 Operational quadrants of electrical motors

Depending on the design of the control unit, electrical drives operate only in the first quadrant (e.g. umps) or in all four quadrants (eg. hoists).

2.1.3

Classification according to technical realisation

.1.3 Classification according to technical realisation Electrical motors are offered in various designs, each with specific strengths and weaknesses as well s preferred power ranges. Taking into account the different characteristics of control units, a large Electrical motors are offered in various designs, each with specific umber of drive technologies are produced.

strengths and weaknes-

ses as well as preferred power ranges. Taking into account the different characteristics of control units, a large number of motor technologies are produced. 6

According to the shape of the motor current, DC motors and 1-phase or 3-phase AC-motors are distinguished: DC motors use a DC motor. For smaller motors, the required magnetic field is generated with permanent magnets. These DC motors can be operated both at a constant DC voltage as well as at pulse converters. AC motors use motors that are operated with 1-phase or 3-phase AC current. The frequency of the motor current has a decisive influence on the motor speed. Synchronous motors follow in their rotary motion exactly the frequency of the feeding current. They are always operated with a control unit. In practice, synchronous motors of smaller design are often referred to a brushless DC motors. Depending on the shape of the current flowing in the motor windings, synchronous motors are divided in motors with block commutation and motors with sinusoidal commutation.

Selection of the optimum motor technology At asynchronous motors a difference between the frequency of the motor current and the rotational frequency occurs. They can be operated either directly on the AC grid as well as on actuators. If a motor is operated on a control unit, the control unit decides on the possibility and quality of the speed adjustability. Simple control units such as frequency converters or commutation electronics enable speed changes within the range of seconds. High-performance control units (e.g. servo controllers) adjust the motor position and speed very precisely within a few milliseconds.

Figure Figure4 4Motor Drivetechnologies technologies

Corresponding to the shape of the motor current, DC drives and 1-phase or 3-phase AC-drives are distinguished 2.2 â&#x20AC;˘ DCSelection the optimum motor technology drives use aofDC motor. For smaller motors, the required magnetic field is generated with permanent magnets. These DC motors can be operated both at a constant DC voltage as well as at pulse From the theoretically available motor technologies, the optimum motor technology has to converters. be selected and subsequently â&#x20AC;˘ AC drives use motors thatdimensioned. are operated with 1-phase or 3-phase AC current. The frequency of the motor current has a decisive influence on the motor speed. o Synchronous motors follow in their rotary motion exactly the frequency of the feeding current. They are always operated with a control unit. In practice, synchronous motors of smaller design are often referred to as brushless DC motors. Depending on the shape of the current flowing in the motor windings, synchronous motors are divided in

Selection of the optimum motor technology The suitable motor technology can easily be selected by using table 2 which contains all motor technologies that may be considered. The form allows the selection in two steps: In the first step, all the motor technologies which cannot meet the technical requirements of the application are eliminated. Technical exclusion criteria are used for this purpose. If a motor technology can not cover these criteria, it is excluded for the second step. In the upper part of Table 2 for each exclusion criterion it is marked, if the motor technology meets the criterion (table field marked with an "x"), or if it doesnâ&#x20AC;&#x2122;t (table field is empty). In the second step, the remaining motor technologies are compared with each other and the optimum solution is determined. Performance criteria are used for this. In the lower part of Table 2, for each performance criterion it is indicated, how the motor technology satisfies the criterion (table field is filled with a corresponding performance number). The following exclusion criteria are used: Application The applications according to chapter 1 (constant speed motor, variable speed motor, servo motor) cannot be implemented with all motor technologies. Power range Motor technologies are only available in certain power range. If the application requires a different power range, the motor technology cannot be used. Start-up If the motor starts up against a high load, some motor technologies are not applicable and have to be deselected. Power supply Constant speed motors in particular can only be operated with the appropriate power supply. Depending on the available power supply network certain motor technologies have to be excluded. To exclude technical unsuitable motor technologies select in the upper part of Table 2 in the second column the criteria valid for the application. Then, go to the right in each selected row and deselect all columns that are not marked with "x". The motor technologies in the deselected columns do not have to be considered for the second selection step. The following performance criteria are used: Small motor size Depending on the type of motor, different sizes can be obtained with comparable power.

Selection of the optimum motor technology High speed accuracy and high dynamics Depending on the design of the control unit, the motor technologies differs in their ability to set the speed precisely and dynamically. Low grid perturbation Depending on the presence of a control device, the motor technologies load the power supply network during start-up with a high or low start-up current. Low cost The motor technologies have different costs because of the different types of motor and control unit. Low complexity Depending on the design, the motor technologies are more or less complex and create different requirements on the skills of the maintenance stuff. Long lifetime, low maintenance efforts Depending on the motor type, the motor technology is low-maintenance and has a long lifetime. To select the optimum motor technology you define the priorities of the performance criteria for the application under consideration in the lower part of Table 2 in the second column. If the criterion is very important to you, set the value to 10. If the criterion is unimportant, enter 0 for the priority. Then, move to the right in each selected row and multiply the priority by the performance number and enter the resulting score into the corresponding field. If, for all technically suitable motor technologies, the scores for all performance criteria are determined, add these columns to a total score. The motor technology with the highest overall score is the optimum motor technology that should be selected for the application under consideration.

Brushless DC motor (BG)with power electronics and open loop V-Control

Stepper motor with control device

Reluctance motor on frequency converterwith V/f-Control

External excited DC motorwith power converter

Permanent magent DC motor with pulse converter and closed loop control (GR)

Permanent magent DC motor with pulse converter and open loop voltage control (GR)

Permanent magent DC motor (GR) on DC grid

Motor technology covers exclusion criterion

Shunt-wound DC motor on DC gridwith start-up resistor

Motor technology does not cover exclusion criterion

Shunt-wound DC motor on DC grid

Available motor technologies

Series-wound DC motor on DC grid

Selection of the optimum motor technology

Application x

Constant speed motor Variable speed motor

x x

Servo motor

Exclusion criteria

Selection of the required exclusion criteria Choose one for each criterion.

Power rangeGrid < 100 W

< 1 kW

x x

< 50 kW

< 500 kW

> 500 kW

Start-up Without or with low load

With load

DC-grid

1-phase AC grid

3-phase AC grid

Grid x

Suitability Features

Performance criteria

Small size

High speed accuracy and high dynamics

Low grid disturbances

Low cost

Low complexity

Long life time, low maintenance efforts

Ranking Table 2 Form for selecting the optimum motor technology

Enter product from priority and performance number

Priority of performance criteria to0 0 0: not 0 important 1 2 10: very important

Enter the sum of the points 0

Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

x x

x x x

x x x x

x x

5Performance 5

number

x x x x

x x x

x Capacitor motor on AC grid Shaded pole motor on AC grid Universal motor on AC grid 3-phase asynchronous motor on AC grid (DR) 3-phase asynchronous motor on AC gridwith soft starter (DR) 3-phase asynchronous motor on AC gridwith star-delta start-up (DR) 3-phase asynchronous motor on frequency converter with V/f-Control 3-phase asynchronous motor on frequency converter with vector control 3-phase asynchronous motor with servo frequency converter 3-phase asynchronous motor with slip ringson AC grid with resistor start-up

External excited synchronous motor with sinusoidal commuation and frequency converter

Brushless DC motor (BG)with integrated power electronics dMove

Brushless DC motor with servo amplifier (BGE/DME)

Selection of the optimum motor technology

x x x x x x x x x x

x x

x x x

x x

x x x

x x

x x x

x x

x x x

x x

x x x

x x

x x x x

x x x

x x x x x x x

Continuation Table 2

Selection of the optimum motor technology

2.3 2.3.1

Electrical motors from Dunkermotoren Permanent Magnet DC Motors - Series GR

Brush type commutated DC motors from Dunkermotoren are notable for robust and maintenance-free design, longer lifetime than commutated motors from other manufacturers, motor insulation class E, extremely high short time overload capacity of the motor and especially high quality due to fully automated production lines. The DC motors can be combined with control electronics, gearboxes, brakes and encoders in a modular system to provide flexible, adaptable, market-oriented solutions.

2.3.2

Brushless DC Motors - Series BG

Brushless motors (EC motors) from Dunkermotoren are notable for very high lifetime, high efficiency, dynamic acceleration, robust design, high overload capability and protection class up to IP65. Combined in a modular system with control gearboxes, brakes and encoders, these electronically commutated DC motors provide a flexible, adaptable, market-oriented solution. Following electronic variants are available: » dCore integrated hall sensors » dGo

» dPro Integrated sinusoidal servo controller

integrated commutation

and integrated high resolution

electronics

encoders for speed- positioning-

» dMove integrated controller for speed-

and current control operation,

control through field bus, industrial

control through field bus or I/Os

Ethernet, I/Os or stand-alone

operation

Selection of the optimum motor technology

2.3.3

Asynchronous AC Motors - Series KD/ DR

Dunkermotoren offers single-phase AC motors of the series KD and three-phase AC motors of the series DR. The motors can be combined with brakes and gearboxes.

2.3.4

Linear Systems - Series ST | CASM | LS

Dunkermotoren offers a wide spectrum of linear motor technology. Whether direct tubular linear motors for highly dynamic positioning tasks or spindle systems with thrust forces in the range of kilo Newton.

2.4

Example: Pump motor

The form for selecting the optimum motor technology is applied to a pump motor. The following requirements are specified for the pump motor: Operating conditions: Series application in heating systems, long lifetime, maintenance-free,

easy handling by service technicians

Power supply:

DC 24 V

Power range:

Approx. 50 W

Speed adjustment: Not required In the first step, the technically suitable motor technologies are extracted. According to table 5, the following motor technologies are usable: • Series wound DC motor on DC grid • Permanent magnet DC motor (GR) on DC grid • Brushless DC motor (BG) with power electronics and open loop V-Control • Brushless DC motor (BG) with integrated power electronics dMove • Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

Selection of the optimum motor technology For the second step, the requirements of the application are weighted and the priorities are assigned: Performance criterion

Priority

Reason

Small size of the motor

This is of little importance for this application.

High speed accuracy and high dynamics

Accuracy and dynamics are not very important for pumps in general.

Low grid perturbation

Since the power is relatively small, grid perturbation due to start-up processes are hardly to be expected.

Low cost

Since it is a standard application, low costs for motor technology are very important.

Low complexity

Easy handling is expressly required.

Long lifetime, low maintenance efforts

Maintenance-free operation is expressly required.

Table 3 Prioritisation of the performance criteria for selecting the optimum motor technology for a pump motor

This results in the following scores for the suitable motor technologies: Motor technology

Score

Series-wound DC motor on DC grid

Permanent magnet DC motor (GR) on DC grid

104

Brushless DC motor (BG) with power electronics and open loop V-Control

135

Brushless DC motor (BG) with integrated power electronics dMove

110

Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

Table 4 Ranking of motor technologies for a pump motor

The optimum motor technology is thus the brushless DC motor (BG) with power electronics and open loop V-Control.

2.5

Example: Door motor

The form for selecting the optimum motor technology is subsequently applied to a motor for an elevator door. The following basic conditions are given for the door motor: The form for selecting the optimum motor technology is subsequently applied to a motor for an elevator door. The following basic conditions are given for the door motor: Operating conditions: Series application, elevator doors in the living area, small installation space for engine and gearbox required, maintenance-free, handling by trained service technicians

Selection of the optimum motor technology Power supply:

DC 24 V

Power range:

Approx. 200 W

Speed adjustment:

Positioning function required

In the first step, the technically suitable motor technologies are extracted. According to Table 8, the following motor technologies are usable: • Permanent magnet DC motor with pulse converter and closed loop control (GR) • Brushless DC motor with servo amplifier (BGE/DME) • Brushless DC motor (BG) with integrated power electronics dMove • Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro • 3-phase asynchronous motor with servo frequency converter For the second step, the requirements of the application are weighted and the priorities are assigned: Performance criterion

Priority

Reason

Small size of the motor

The small size is important.

High speed accuracy and high dynamics

Accuracy and dynamics are very important for positioning motors.

Low grid perturbation

Grid perturbation due to start-up is less relevant.

Low cost

Since it is a series application, low costs for the motor technology are very important

Low complexity

Handling by trained service technicians.

Long lifetime, low maintenance efforts

Maintenance-free operation is required.

Table 6 Prioritisation of the performance criteria for selecting the optimum motor technology for a door motor

This results in the following scores for the suitable motor technologies: Motor technology

Score

Permanent magnet DC motor with pulse converter and closed loop control (GR)

Brushless DC motor with servo amplifier (BGE/DME)

Brushless DC motor (BG) with integrated power electronics dMove

Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

3-phase asynchronous motor with servo frequency converter

Table 7 Ranking of motor technologies for a door motor

Stepper motor with control device

Reluctance motor on frequency converterwith V/f-Control

Permanent magent DC motor (GR) on DC grid

External excited DC motorwith power converter

Shunt-wound DC motor on DC gridwith start-up resistor

Permanent magent DC motor with pulse converter and closed loop control (GR)

Shunt-wound DC motor on DC grid

Permanent magent DC motor with pulse converter and open loop voltage control (GR)

Series-wound DC motor on DC grid

Selection of the optimum motor technology

Application x

Constant speed motor Variable speed motor Servo motor

Power range

Exclusion criteria

< 100 W

< 1 kW

x x

< 50 kW

< 500 kW

> 500 kW

Start-up x

Without or with low load

With load

Energieversorgungsnetz x

DC-grid

1-phase AC grid

3-phase AC grid

Suitability

Performance criteria

Features 1

Small size

High speed accuracy and high dynamics

Low grid disturbances

Low cost

Low complexity

Long life time, low maintenance efforts

Ranking

104

Table 5 Form for selecting the optimum motor technology for a pump motor

x x x

135 x

x x x x x

x x x x

Capacitor motor on AC grid Shaded pole motor on AC grid Universal motor on AC grid 3-phase asynchronous motor on AC grid (DR) 3-phase asynchronous motor on AC gridwith soft starter (DR) 3-phase asynchronous motor on AC gridwith star-delta start-up (DR) 3-phase asynchronous motor on frequency converter with V/f-Control 3-phase asynchronous motor on frequency converter with vector control 3-phase asynchronous motor with servo frequency converter 3-phase asynchronous motor with slip ringson AC grid with resistor start-up

External excited synchronous motor with sinusoidal commuation and frequency converter

Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

x Brushless DC motor (BG)with integrated power electronics dMove

x Brushless DC motor with servo amplifier (BGE/DME)

Brushless DC motor (BG)with power electronics and open loop V-Control

Selection of the optimum motor technology

x x x x x x x x x x

x x x x x

x x x

x x x x x x x x

x x x x x x x

x x x x x

x x

x x x x

x x x

x x

x x x x x x

x x

110

Continuation Table 5

Stepper motor with control device

Reluctance motor on frequency converterwith V/f-Control

Permanent magent DC motor (GR) on DC grid

External excited DC motorwith power converter

Shunt-wound DC motor on DC gridwith start-up resistor

Permanent magent DC motor with pulse converter and closed loop control (GR)

Shunt-wound DC motor on DC grid

Permanent magent DC motor with pulse converter and open loop voltage control (GR)

Series-wound DC motor on DC grid

Selection of the optimum motor technology

Application Constant speed motor Variable speed motor x

Servo motor

Power range < 100 W Exclusion criteria

< 1 kW

x x

< 50 kW

< 500 kW

> 500 kW

Start-up

Without or with low load

With load

Grid x

DC-grid

1-phase AC grid

3-phase AC grid

Suitability

Performance criteria

Features 5

Small size

High speed accuracy and high dynamics

Low grid disturbances

Low cost

Low complexity

Long life time, low maintenance efforts

Ranking Table 8 Form for selecting the optimum motor technology for a door motor

x x x

x x

x x x x x

x x x x

External excited synchronous motor with sinusoidal commuation and frequency converter

Permanent magnet synchronous motor with sinusoidal commutation (BG) and integrated servo amplifier dPro

x Brushless DC motor (BG)with integrated power electronics dMove

x Brushless DC motor with servo amplifier (BGE/DME)

Brushless DC motor (BG)with power electronics and open loop V-Control

Selection of the optimum motor technology

x x x x x x x x x x

x x x x x

x x x

x x x x x x x x

x x x x x x x

x x x x x

x x

x x x x

x x x

x x x x

x x x

x x

x x x x x x

x x

Continuation Table 8

Description of the load cycle

3.1

Kinematic equations

The load cycle of an application is described by the chronological sequence of the motion and force variables. If the movement is cyclically, only one period is considered. 3 Beschreibung des Lastspiels Kinematische 3 3.1Beschreibung desGleichungen Lastspiels Das Kinematische Lastspiel einer durch den between zeitlichen translational Verlauf der Bewegungsund movements. KraftgrĂśĂ&#x;en 3.1Mathematically Gleichungen it isAnwendung necessary wird to distinguish and rotational beschrieben. Ist der Bewegungsablauf zyklisch, wird nur ein Zyklusder des Lastspiels betrachtet. Das Lastspiel einer Anwendung wird durch den zeitlichen Verlauf Bewegungsund KraftgrĂśĂ&#x;en As a matterIst inist fact, this distinction iszyklisch, oftenund not made. In particular the terms position (for distanMathematisch zwischen translatorischen rotatorischen Bewegungen zu unterscheiden. beschrieben. der Bewegungsablauf wird nur ein Zyklus des Lastspiels betrachtet. Umgangssprachlich wirdacceleration diese Unterscheidung oft nicht getroffen. Insbesondere die Begriffe Position Mathematisch ist speed, zwischen translatorischen undare rotatorischen Bewegungen zu unterscheiden. ce and angle), and jerk used for both types of movement. (fĂźr Weg und Winkel), Beschleunigung und Ruck Insbesondere werden fĂźr beide Umgangssprachlich wirdGeschwindigkeit, diese Unterscheidung oft nicht getroffen. die Begriffe Position (fĂźrBewegungsformen Weg und Winkel),verwendet. Geschwindigkeit, Beschleunigung und Ruck werden fĂźr beide The movement ofverwendet. bodies is described by the following kinematic equations. Bewegungsformen Die Bewegung von KĂśrpern wird durch die folgenden kinematischen Gleichungen beschrieben.

Die Bewegung von KĂśrpern wird durch die folgenden kinematischen Gleichungen beschrieben. Translation Rotation Translation Rotation Angle Weg s = f(t) đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą) s: Distance Ď&#x2022;: Winkel s = f(t) đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą) s: Weg Ď&#x2022;: Winkel đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; Angular speed Geschwindigkeit v: Velocity Ď&#x2030;: Winkelgeschwindigkeit đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł =đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą v: Geschwindigkeit Ď&#x2030;: Winkelgeschwindigkeit đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł = đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; =đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą Speed (1/s) đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; n: Drehzahl đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; =2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; n: Drehzahl 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; Beschleunigung Angular acceleration a: Acceleration Îą: Winkelbeschleunigung đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; =đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą a: Beschleunigung Îą: Winkelbeschleunigung đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą Ruck Jerk Angular jerk Winkelruck j: Ď : đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014; = đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź j: Ruck Ď : Winkelruck đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014; = đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; = Tabelle 9 Kinematische Gleichungen đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą der Translation und der Rotation đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą Table 9 Kinematic equations of translation and rotation Tabelle 9 Kinematische Gleichungen der Translation und der Rotation

Die Bewegungen der KĂśrper werden durch KrĂ¤fte hervorgerufen, die auf sie einwirken. Der The movementszwischen of the bodies und are caused by forcessich which upon them. The relationship Bewegung aus act dem Newton bzw. DieZusammenhang Bewegungen der KĂśrperKraft werden durch KrĂ¤fteergibt hervorgerufen, dieSatz auf von sie einwirken. Derseiner Erweiterung zum Kraftbzw. Drehmomentgleichgewicht. between forcezwischen and motion results from the set of Newton its extension to the force /or Zusammenhang Kraft und Bewegung ergibt sich ausordem Satz von Newton bzw.and seiner Erweiterung zum Kraft- bzw. Drehmomentgleichgewicht. torque equilibrium.

Abbildung 5 KrĂ¤fte und Drehmomente Figure 5 Forces and torques

Abbildung 5 KrĂ¤fte und Drehmomente Die vom elektrischen Antrieb bereitgestellten KrĂ¤fte und Drehmomente werden â&#x20AC;˘ zum Beschleunigen und Bremsen der mechanischen Elemente und torques provided byder theArbeitsmaschine electrical are used LastkrĂ¤fte DieThe vom elektrischen Antrieb bereitgestellten KrĂ¤fte motor und Drehmomente werden und â&#x20AC;˘forces zumand Kompensieren der von geforderten Lastdrehmomente Beschleunigen und Bremsen dermechanical mechanischen Elemente und â&#x20AC;˘â&#x20AC;˘ forzum acceleration and deceleration of the elements verwendet. â&#x20AC;˘ zum Kompensieren der von der Arbeitsmaschine geforderten LastkrĂ¤fte und â&#x20AC;˘ forLastdrehmomente compensation of the load forces and load torques required by the working machine. verwendet. Translation 22

Translation Kraft F:

Rotation

Rotation M: Drehmoment

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

zum Beschleunigen und Bremsen der mechanischen Elemente und zum Kompensieren der von der Arbeitsmaschine geforderten LastkrĂ¤fte und Lastdrehmomente Description of the load cycle verwendet. â&#x20AC;˘ â&#x20AC;˘

Translation F:

Force Kraft

Force equilibrium Kraftgleichgewicht

Beschleunigungskraft FB: Acceleration force

Rotation

đ??šđ??šđ??šđ??š = đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ + đ??šđ??šđ??šđ??šđ??żđ??żđ??żđ??ż đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

Torque Drehmoment

Torque equilibrium Drehmomentgleichgewicht

MB: Beschleunigungsmoment Acceleration torque

Lastkraft FL: Load force J: Lastdrehmoment ML: Load torque Masse der sich linear bewegenden m: Mass of linear moving mechanical elements d: mechanischen Elemente Tabelle 10 Kraft- und Drehmomentgleichgewicht

Table 10 Force and torque equilibrium

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ??ľđ??ľđ??ľđ??ľ + đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ??żđ??żđ??żđ??ż đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ??ľđ??ľđ??ľđ??ľ = đ??˝đ??˝đ??˝đ??˝

Moment of inertia of rotating mechanical TrĂ¤gheitsmoment der rotierenden elements mechanischen Elemente Diameter at which the load force engages Durchmesser, an dem die Lastkraft angreift

In order to be able to use the force and torque equations, â&#x20AC;˘ the movement sequence, â&#x20AC;˘ the masses and moments of inertia to be moved and â&#x20AC;˘ the applied load forces and load torques have to be known. Their investigation is the subject of the following chapters.

3.2 3.2.1

Motion laws Motion laws for constant speed motors

The motion of constant speed motors is determined by the On and Off commands which the motor receives from the superimposed controller or directly from the operator. At the beginning of each change in state, a start-up or a ramp-down occurs. During the (continuous) operation, the operating point is determined by the load. The speed of the motor depends on the load. There is no active influence on the motion variables. The power the motor has to apply during continuous operation is often used as selection criteria. The duration of the entire load cycle for constant speed motors is practical in the range of minutes or hours. The distances or revolutions the motors move forward are theoretically unlimited. Constant speed motors are therefore designed with rotary motors.

3.2.2

Motion laws for variable speed motors

The motion of variable speed motors is defined by the time profile of the speed. Speed changes are carried out with a maximum permissible acceleration. By slow changes of the speed, for example, the tilting of objects on conveyor belts and vehicles, the tearing of webs, sudden pressure changes in pipelines shall be prevented.

3.2 Bewegungsgesetze 3.2.1 Description Bewegungsgesetze für Konstantantriebe 3.2.2 Bewegungsgesetze für drehzahlveränderlich of the load cycle Der Bewegungsablauf von Konstantantrieben durch die Ein-drehzahlveränderlicher und Ausschaltbefehle, die dw Derwird Bewegungsablauf Antriebe Drehzahländerungen werden mit einer Antrieb von der überlagerten Steuerung oderdefiniert. direkt vom Bediener erhält, bestimmt. Zumaxima Begin In the case of variable speed the acceleration and deceleration processes slow. Sosollen the beispi Durch langsame Änderungen derare Drehzahl Zustandsänderung kommt es motors, zu einem Hochoder einem Rücklauf. Während des (Dauer-)Be •verydas Kippen vononGegenständen auf Förderbände acceleration of the mechanical has only aDie small influence the required wird der Arbeitspunkt durch die elements Last bestimmt. Drehzahl des Motors stelltmotor sich lastabhäng • das Reißen von Warenbahnen, Eine aktive Beeinflussung der Bewegungsgrößen erfolgt nicht. torque. The duration of the entire load cycle is practical in the range of minutes or hours. The dis• plötzliche Druckänderungen in Flüssigkeitsleitun Als Kenngröße für die Auslegung wirdforward oft dieare Leistung, dieunlimited. der Motor im Dauerbetrieb tances or revolutions the motors move theoretically Therefore variable speed aufbrin verhindert werden. muss, motors verwendet. are designed with rotary motors.

Abbildung 6 Drehzahlverlauf einesAb Konstantantriebs bei konstanter Las dre

On/Off

Figure 6 Speed profile of a constant speed motor at constant load

Figure 7 Speed profile of a variable speed motor

Bei drehzahlveränderlichen Antrieben verlaufen die Hoc

Die Zeitdauer des gesamten Lastspiels liegtdie fürBeschleunigung Konstantantriebe praktisch im Elemente Bereich von M der mechanischen nur ein oder Stunden. Die Wege Umdrehungen, die die Antriebe dabei zurücklegen, sind theore Motordrehmoment hat. Die Zeitdauer des g 3.2.3 Motion lawsbzw. for servo motors erforderliche von Rotationsmotoren Minuten oder Stunden. Die Wege bzw. Umdrehunge unbegrenzt. Konstantantriebe sind deshalb mit ausgeführt.

theoretisch unbegrenzt. Drehzahlveränderliche Antriebe

For servo motors, the position is the leading ausgeführt. value. The application requires a certain distance

move and specifies the permissible positioning time for this movement. Using a motion law, 3.2.2 to Bewegungsgesetze für drehzahlveränderliche Antriebe the velocity and the drehzahlveränderlicher acceleration are derived from this. Der Bewegungsablauf Antriebe wird über den zeitlichen Verlauf der D definiert. Drehzahländerungen werden mit einer maximal zulässigen Beschleunigung ausgefü 3.2.3 Bewegungsgesetze für Servoantriebe Bei Servoantrieben ist dieare Position die führende Größe. Aus der Applikation werden im and at the end of the movement laws defined forDrehzahl a positioning process. At heraus the beginning Durch The langsame Änderungen der sollen beispielsweise 17 Allgemeinen der Verfahrweg und die dafür zulässige Verfahrzeit vorgegeben. Unter Anwendung eines • movement das Kippen von Förderbändern und Fahrzeugen, the motor isGegenständen at standstill. Bewegungsgesetzes werden daraus der Verlauf der auf Geschwindigkeit und der Beschleunigung • abgeleitet. das Reißen von Warenbahnen, Die Bewegungsgesetze werden für einen Positioniervorgang definiert. Der Antrieb befindet sich zu und am Ende des Positioniervorgangs im • Beginn plötzliche inStillstand. Flüssigkeitsleitungen Motion lawsDruckänderungen and parameters verhindert werden. und Kenngrößen Bewegungsgesetz

The motion law used determines the specific mathematical relationship between position,

Abbildung 7 Drehzahlverlauf eines drehzahlveränderlichen Antriebs

velocity, acceleration and jerk. Das verwendete Bewegungsgesetz bestimmt die konkreten mathematischen All movement variables are interrelated Zusammenhänge zwischen Position, Geschwindigkeit, Beschleunigung influence each other.und Ruck. Alle Bewegungsgrößen stehen miteinander in Zusammenhang und beeinflussen sich gegenseitig.

and

Bewegungsgrößen, die den Positioniervorgang beschreiben

TPos: 11 Important Positionierdauer, Zeitdauer des gesamten Table parameters for motion laws Positioniervorgangs sPos, ϕ Pos:

Positionsänderung, Sollwert des Weges bzw. Winkels

während des gesamten Positioniervorgangs vmax, ω max: Maximale Geschwindigkeit Bei drehzahlveränderlichen Antrieben verlaufen die Hoch- und Rücklaufvorgänge so langsam amax, α max: Maximale Beschleunigung während des gesamten Positioniervorgangs die Beschleunigung der mechanischen Elemente nur einen sehr geringen Einfluss auf das Maximaler Ruck während des gesamten Positioniervorgangs jmax, ρ max: Tabelle 11 Wichtige Kenngrößen für Bewegungsgesetze erforderliche Motordrehmoment hat. Die Zeitdauer des gesamten Lastspiels liegt praktisch im 24 Minuten oder Stunden. Die Wege bzw. Umdrehungen, die die Antriebe dabei zurücklege von

Das verwendete Bewegungsgesetz bestimmt die konkreten mathematischen Description of the load cycle Zusammenhänge zwischen Position, Das verwendete Bewegungsgesetz Das verwendete Bewegungsgesetz Geschwindigkeit, und Ruck. bestimmt dieBeschleunigung konkreten mathematischen bestimmt die konkreten mathematischen Alle Bewegungsgrößen stehen miteinander in Zusammenhänge Position, Zusammenhänge zwischen zwischen Position, Zusammenhang und beeinflussen sichund Ruck. Geschwindigkeit, Beschleunigung Geschwindigkeit, Beschleunigung und Ruck. Parameters describing the positioning process Alle Bewegungsgrößen stehen miteinander in Alle gegenseitig. Bewegungsgrößen stehen miteinander in Zusammenhang und beeinflussen sich Zusammenhang und beeinflussen sich

TPos: gegenseitig. process Positioning time, duration of the entire positioning gegenseitig. sPos, φ Pos: Position setpoint, distance or angle to travel during the entire positioning process vmax, ωmax: Maximum velocity or angular speed during the entire positioning process amax , αmax: Maximum acceleration or angularbeschreiben acceleration during the entire positioning process Bewegungsgrößen, die den Positioniervorgang jmax TPos, :ρmax: Bewegungsgrößen, Positionierdauer, des gesamten Positioniervorgangs Maximum jerkZeitdauer or angular jerk during entire positioning process die den Positioniervorgangthe beschreiben

Bewegungsgrößen, die den Positioniervorgang beschreiben sPos, ϕ Pos: TPos:Positionsänderung, Sollwert des Weges bzw. Winkels Positionierdauer, Zeitdauer desPositioniervorgangs gesamten Positioniervorgangs TPos: Positionierdauer, Zeitdauer des gesamten ,ω MaximalePositionsänderung, Geschwindigkeit desWeges gesamten Sollwert des bzw. Positioniervorgangs Winkels Pos, ϕ Pos: sPosv, max ϕ Pos : max:sPositionsänderung, Sollwert deswährend Weges bzw. Winkels , α : Maximale Beschleunigung während des gesamten Positioniervorgangs a Maximale Geschwindigkeit während desPositioniervorgangs gesamten Positioniervorgangs max, ω max: Geschwindigkeit ω max:max vMaximale während des gesamten vmax,max ρ max Maximaler Ruckare während des gesamten Positioniervorgangs max , α max: Beschleunigung Maximale Beschleunigung während desPositioniervorgangs gesamten Positioniervorgangs max The motion laws considered. , α,max : : aMaximale während des gesamten amaxjfollowing , ρ max: Ruck Maximaler Ruck desPositioniervorgangs gesamten Positioniervorgangs max Tabelle Wichtige Kenngrößen fürwährend Bewegungsgesetze , ρ max: 11jMaximaler während des gesamten jmax Tabelle 11 Wichtige Kenngrößen für Bewegungsgesetze Tabelle 11 Wichtige Kenngrößen für Bewegungsgesetze

Time-optimised

Loss-optimised

Stellvertretend werden folgende Bewegungsgesetze betrachtet.

Sine function

Stellvertretend werden folgende Bewegungsgesetze Stellvertretend werden folgende Bewegungsgesetze betrachtet.betrachtet.

Reduces losses during positi-Sinus Protects the mechanics. Allows the shortest positioGeschwindigkeitsoptimal Verlustoptimal Geschwindigkeitsoptimal Verlustoptimal Sinus Geschwindigkeitsoptimal Verlustoptimal Ermöglicht die kürzeste Senkt die Verluste beim The Sinus Schont dievelocity Mechanik. ning time with a maximum oning to a minimum. top The or speed profile Ermöglicht die kürzeste Senkt die Verluste beim Schont die Ermöglicht die kürzeste Senkt die Verlusteauf beim Mechanik. Positionierdauer bei einem Positionieren ein Minimum Schont Derdie Verlauf der Mechanik. Geschwindigkeit Positionierdauer bei einem Positionieren auf ein Minimum Der Verlauf der auf, Geschwindigkeit Positionierdauer bei einem Positionieren auf ein Minimum Der Verlauf der Geschwindigkeit available velocity or speed achieved weisthas noKnicke kinks, torque surges maximal torque. verfügbaren ab. Die erreichte keine maximal verfügbaren ab. Die erreichte ab. Die erreichte weist keineauf, Knicke auf, maximal verfügbaren keine Knicke Drehmoment. Spitzengeschwindigkeit ist am weistDrehmomentstöße werden is lowest compared to other are avoided. Drehmoment. Spitzengeschwindigkeit ist am Drehmomentstöße Drehmoment. Spitzengeschwindigkeit ist am Drehmomentstöße werden werden niedrigsten im Vergleich zu vermieden. im zu Vergleich zu vermieden. niedrigstenniedrigsten im Vergleich vermieden. motion laws.Bewegungsgesetzen. anderen Bewegungsgesetzen. anderen anderen Bewegungsgesetzen.

Tabelle 12 Bewegungsgesetze Tabelle 12 12 Bewegungsgesetze Tabelle Bewegungsgesetze Table 12 Motion laws

The following procedure is recommended for the definition of the motion laws' parameters at a given position setpoint: 18

1. Depending on the requirements of the application according to high dynamics, low losses and low 18 stress on the mechanics the optimum motion law is selected. 2. Derivation of vmax respectively ωmax and amax respectively αmax for the required position setpoint and positioning time. 3. Determination of the maximum speed and acceleration needed and comparison with the limit values vP respectively ωP and aP respectively αP permitted by the application 4. If the maximum permissible acceleration is exceeded, set amax = aP respectively αmax = αP and determine TPos, vmax respectively ωmax. 5. If the maximum permissible speed is exceeded, set amax = aP, vmax = vP respectively αmax = αP, ωmax = ωP and insert a constant speed phase with v = vmax and determine TPos The following diagrams show the mathematical relationships for the individual motion laws.

SegSegSegment ment ment

Segment

SegSegment ment SegSeg-SegSegSegSegSegment ment ment ment ment ment ment

slation, Vorgabewerte: sPos , TPos Translation, Vorgabewerte: sPosDiese , TPos Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwen Diese kommen bei translatorischen Bewegungen zur Anwendung Anwendung ranslation, Vorgabewerte: sTPos ,Bewegungsgesetze TPos ndungVorgabewerte: Description of the cycle ation, sPosload ,Positionierzeit Überschreitung zulässigen Geschwindigkeit und der zul PosPositionierzeit ohne ohne Überschreitung zu zulässigen Geschwindigkeit und der zulässig Diese Bewegungsgesetze kommen beizutranslatorischen Bewegungen zur Anwen Anwendung ranslation, Vorgabewerte: sPosDiese ,Positionierzeit TPos Bewegungsgesetze kommen bei zu translatorischen Bewegungen zur ohne Überschreitung zulässigen Geschwindigkeit undAnwend der zul Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, nwendung anslation, Vorgabewerte: sPos ,Parameter T Pos Translation, Vorgabewerte: sPos ,T Pos dung Positionierzeit ohne Überschreitung zutranslatorischen zulässigen Geschwindigkeit und der zulä Bewegungsgesetz Weg G These motion lawsPositionierzeit are used inBewegungsgesetze translatory movements if desired position canBewegungen be realized gungsgesetz Parameter Weg Gesch ohne Überschreitung zuthe zulässigen Geschwindigkeit und der zulässigen Diese kommen bei zur Anwen ranslation, Vorgabewerte: , T Pos n, Vorgabewerte: sPos , TPossPos nwendung Diese Bewegungsgesetze kommen translatorischen Bewegungen zur Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zurAnwend Anwen Bewegungsgesetz Parameter Weg G Positionierzeit ohneexceeding Überschreitung zu zulässigen Geschwindigkeit und der zulä time without thebei permissible velocity and the permisnwendungwithin the required positioning Anwendung Positionierzeit ohne Überschreitung zuzu zulässigen Geschwindigkeit und der zulä ewegungsgesetz Parameter Weg Ge Positionierzeit ohne Überschreitung zulässigen Geschwindigkeit und der zul Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwen Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wen ungsgesetz Parameter Weg Geschw sible acceleration. Anwendung ng 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ohne Überschreitung 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡und 2 Translation, Vorgabewerte: sParameter , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 TPos Positionierzeit zu𝑠𝑠𝑠𝑠 zulässigen und der zul Pos Positionierzeit ohne zulässigen Geschwindigkeit Be 𝑎𝑎𝑎𝑎= I zu ewegungsgesetz Weg Ge 2 I 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 = 4Überschreitung = 2𝑠𝑠𝑠𝑠 = 22𝑇𝑇𝑇𝑇∙Geschwindigkeit 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝐼𝐼𝐼𝐼 der zulässigen Translation, Vorgabewerte: sPos , TPos𝑇𝑇𝑇𝑇 2𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ewegungsgesetz Parameter Weg Ge Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Positio 2 Bewegungsgesetz Parameter Weg G 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 Translation, Vorgabewerte: sPos , 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 TPos = 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 Anwendung I beizutranslatorischen 𝐼𝐼𝐼𝐼 Beschleunigung 𝑠𝑠𝑠𝑠ohne 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 kommen 2 Translation, settings: SPos , T𝑠𝑠𝑠𝑠Bewegungsgesetze Positionierzeit zulässigen Geschwindigkeit und der zulässigen realisiert w Diese Bewegungen zur Anwendung, die gewünschte Position 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Überschreitung Pos 2 wenn 𝑠𝑠𝑠𝑠= Anwendung Parameter 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃4 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Überschreitung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter Weg G 𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡 gsgesetz Anwendung Weg Geschwind Diese Bewegungsgesetze kommen beizu translatorischen Bewegungen zur der Anwendung, wenn die gewünschte Position 22 I Positionierzeit ohne zulässigen Geschwindigkeit zulässigen Beschleunigung realisiert w𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =Positionierzeit 4 𝑠𝑠𝑠𝑠 = 2 ∙ 𝑡𝑡𝑡𝑡 I 𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠 2 𝑇𝑇𝑇𝑇 Überschreitung zu zulässigen Geschwindigkeit und 𝑠𝑠𝑠𝑠 der zulässigen Beschleunigung realisiert we 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Weg SegGeschwindigkeit Bes 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 ohne 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Motion law Parameter 𝑎𝑎𝑎𝑎Parameter 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠∙−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I ment 𝑠𝑠𝑠𝑠Distance 𝑠𝑠𝑠𝑠= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter Weg Besc 2Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑠𝑠𝑠𝑠 = + 𝑡𝑡𝑡𝑡 2 𝑣𝑣𝑣𝑣 2 2 III 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠 = + 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2==4 4 𝑇𝑇𝑇𝑇 Bewegungsgesetz Parameter Besch 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III Weg 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎= 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2Geschwindigkeit 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 II 2𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠== 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 2 ∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2+ 2 2 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 Geschwindigkeitsoptimal 4 4 𝑠𝑠𝑠𝑠I2 𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠2= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 III 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 4 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚== 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 ∙𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2 2𝑠𝑠𝑠𝑠 ∙𝑠𝑠𝑠𝑠= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣 = 4 I I 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 hwindigkeitsoptimal 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 4 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2I 2𝑠𝑠𝑠𝑠= 4 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡 2 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 42𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼+ 2 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠=== ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡2∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑣𝑣𝑣𝑣 = 2𝑣𝑣𝑣𝑣 2 ==4 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2+ ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡=𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 4 2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 I 𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 2 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = + 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 eschwindigkeitsoptimal 𝑣𝑣𝑣𝑣 = 2 III 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III windigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 222∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 + 2 𝑠𝑠𝑠𝑠= ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 −+ 2 𝑠𝑠𝑠𝑠2 2𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −∙ 4 = 22𝑠𝑠𝑠𝑠6 𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡∙ 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 −−2 22𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 IIIIII𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 2 + 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Time-optimised 2−2 =III 2𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 − 2 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 =𝑇𝑇𝑇𝑇2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 4 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 eschwindigkeitsoptimal 2 𝑇𝑇𝑇𝑇 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 + 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃22∙2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑡𝑡𝑡𝑡 − 2 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 − 4 2 𝑣𝑣𝑣𝑣 = 2 𝑎𝑎𝑎𝑎 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = + 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = + 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 2 𝑣𝑣𝑣𝑣 = 2 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 III 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 eschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣 Geschwindigkeitsoptimal 𝑡𝑡𝑡𝑡 2𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠− 2 ∙ 𝑡𝑡𝑡𝑡 3 3 𝑇𝑇𝑇𝑇 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 =𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 6 = 6𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠 = 3𝑠𝑠𝑠𝑠 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃32𝑇𝑇𝑇𝑇∙ 𝑡𝑡𝑡𝑡 2 2−∙ 2 𝑣𝑣𝑣𝑣 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6 3𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 2𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠 Geschwindigkeitsoptimal digkeitsoptimal 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 6 𝑇𝑇𝑇𝑇 3 𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 3 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 2 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 = 6 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 3 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 6𝑇𝑇𝑇𝑇6 𝑠𝑠𝑠𝑠 = 3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 − 23𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 3 ∙ 𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠− = 6∙ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣= = 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 𝑠𝑠𝑠𝑠 3 2 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚== 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 2 2 3 2 2 = 6 3die 𝑇𝑇𝑇𝑇 3 2 22𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ innerhalb 𝑡𝑡𝑡𝑡3 ∙ 𝑡𝑡𝑡𝑡− ∙ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Position 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2wenn 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 3Position 𝑣𝑣𝑣𝑣 =36 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 slatorischen Bewegungen Anwendung, wenn die gewünschte der 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 orischen Bewegungen zur zur Anwendung, gewünschte der geforderten 3 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡2 − innerhalb 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 Verlustoptimal = 𝑠𝑠𝑠𝑠 = 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚der = zulässigen 6 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 == 2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 ∙ 𝑡𝑡𝑡𝑡 36 stoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠Beschleunigung 𝑠𝑠𝑠𝑠 −=2werden 3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎 2𝑠𝑠𝑠𝑠 Beschleunigung 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 werden 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ässigen Geschwindigkeit realisiert 2𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 23 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 igen Geschwindigkeit undund der zulässigen realisiert 𝑠𝑠𝑠𝑠kann. 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝑇𝑇𝑇𝑇∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡3 3 2 kann. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣3 =3 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 = 3 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 22 Verlustoptimal 𝑠𝑠𝑠𝑠 = 3 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 22 33 Loss-optimised 𝑇𝑇𝑇𝑇3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 wenn 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 slatorischen Bewegungen Anwendung, die gewünschte Position innerhalb geforderten 𝑠𝑠𝑠𝑠2wenn orischen Bewegungen zurzur Anwendung, die gewünschte Position innerhalb geforderten 2 der 3 Verlustoptimal 23𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 32𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 erlustoptimal 𝑠𝑠𝑠𝑠3 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 = 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = 3 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 toptimal 2 𝑎𝑎𝑎𝑎Geschwindigkeit = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2kann. 𝑎𝑎𝑎𝑎 2𝜋𝜋𝜋𝜋 = 2𝜋𝜋𝜋𝜋 2 2 3 Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2 ässigen Geschwindigkeit zulässigen Beschleunigung realisiert werden 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 igen Geschwindigkeit undund derder zulässigen Beschleunigung realisiert kann. 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇Beschleunigung 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃werden 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 erlustoptimal 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇3 3 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 en Bewegungen zur Anwendung, wenn Position innerhalb der geforderten 2𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 == 2𝜋𝜋𝜋𝜋gewünschte 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 atorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚die 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚=𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎Beschleunigung erlustoptimal 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣 =2𝜋𝜋𝜋𝜋 Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 realisiert 𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠− innerhalb 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃 eschwindigkeit undAnwendung, der Anwendung, zulässigen werden kann. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 schen Bewegungen zur die gewünschte Position der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ssigen Geschwindigkeit und𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 der zulässigen Beschleunigung werden kann. 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 n Bewegungen zur wenn die gewünschte Position geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2= 𝑇𝑇𝑇𝑇wenn 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 innerhalb 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠22 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 �=𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∙ �𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 �der 𝑠𝑠𝑠𝑠 =realisiert ∙ Beschleunigung �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡�� Geschwindigkeit 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = ∙ − 𝑠𝑠𝑠𝑠 � 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 = 2 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 werden 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 imal 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Translation, 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 nschwindigkeit Geschwindigkeit und der zulässigen Beschleunigung realisiert kann. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Vorgabewerte: s , T 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 und der zulässigen Beschleunigung realisiert werden kann. Pos Pos 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠= ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 = 4 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑠𝑠𝑠𝑠2= 2 2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2Translation, 𝑣𝑣𝑣𝑣Pos= 𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑠𝑠 �2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ �𝑡𝑡𝑡𝑡 −𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣,= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑣𝑣𝑣𝑣 24𝑇𝑇𝑇𝑇= Vorgabewerte: sPos T𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇22𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎Diese =2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 Bewegungsgesetze bei translatorischen Bewegungen zur Anwendung, die gewünschte Positio 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃wenn Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇�𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Translation, Vorgabewerte: s𝑎𝑎𝑎𝑎Pos , TPos Geschwindigkeit Beschleunigung 2 kommen 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Anwendung 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇 Sinus function 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 Sine 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇ohne 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇ohne Positionierzeit zulässigen Geschwindigkeit zulässigen Beschleunigung realisiert w 13 Bewegungsgesetze der Translation Begrenzungen 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠und �𝑠𝑠𝑠𝑠 der 𝑠𝑠𝑠𝑠 zu = ∙ �𝑡𝑡𝑡𝑡 −Bewegungen 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = Diese Bewegungsgesetze kommen bei translatorischen zur Anwendung, wenn die gewünschte Position 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃wenn Anwendung ==2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit 𝑠𝑠𝑠𝑠Überschreitung 𝑠𝑠𝑠𝑠zulässigen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋𝑎𝑎𝑎𝑎der 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Tabelle Geschwindigkeit Beschleunigung Diese kommen beiBeschleunigung translatorischen Bewegungen zur Anwendung, die gewünschte Position 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 Tabelle 13 Bewegungsgesetze Translation ohne 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠Geschwindigkeit 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2Begrenzungen 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠ohne Positionierzeit Überschreitung zu zulässigen und der Beschleunigung realisiert w 2Anwendung 𝑇𝑇𝑇𝑇ohne 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 𝑠𝑠𝑠𝑠Bewegungsgesetze 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Translation Begrenzungen 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 13 Bewegungsgesetze der 𝑣𝑣𝑣𝑣42= ohne 4 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 4 s𝑠𝑠𝑠𝑠2= 2 2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼Tabelle 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 = ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 4 = 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Positionierzeit Überschreitung zu zulässigen Geschwindigkeit und der zulässigen Beschleunigung realisiert we 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 2 2 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Velocity Bewegungsgesetz Parameter Weg Bes Motion law Acceleration 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ohne 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ �𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Bewegungsgesetze 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠== 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣= 𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� ∙ �𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 abelle 13 derParameter Translation 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠Geschwindigkeit Sinus 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠e𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 der Translation 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Begrenzungen 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz WegBegrenzungen Besc 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 13 Bewegungsgesetze ohne 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 = 𝑣𝑣𝑣𝑣 2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 Bewegungsgesetz Parameter Weg Geschwindigkeit Besch 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 − 4 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = −4 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 4 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 4 ∙ 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 2 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠= −4𝑇𝑇𝑇𝑇 4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙2 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎�𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠=− −4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 4 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ �𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 nus 𝐼𝐼𝐼𝐼 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 2 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � = − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 2 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Begrenzungen 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 abelle 132 Bewegungsgesetze der ohne =2𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣Translation 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 2 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2=𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎= = 4 ∙ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼2 𝑣𝑣𝑣𝑣 = 4 𝑠𝑠𝑠𝑠 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼22𝑇𝑇𝑇𝑇13 𝑣𝑣𝑣𝑣der = 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠4Translation = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚4 2 ∙4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 belle Bewegungsgesetze ohne Begrenzungen 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 inus 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 20 =4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2ohne 𝑠𝑠𝑠𝑠Bewegungsgesetze 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣der Translation 𝑇𝑇𝑇𝑇 I 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 13 Begrenzungen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡−𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2− 2 2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 4I 2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 = 2 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎−4 = −4 2 2 𝑣𝑣𝑣𝑣 = 4 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 24 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 = =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2==24 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 nus Sinus 20 13 Bewegungsgesetze der Translation ohne Begrenzungen 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇belle 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃1313 𝑠𝑠𝑠𝑠2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Begrenzungen 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣 = 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Sinus belle Bewegungsgesetze Translation ohne abelle Bewegungsgesetze der 𝑠𝑠𝑠𝑠Begrenzungen = 𝑠𝑠𝑠𝑠 + 2 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2 𝑠𝑠𝑠𝑠 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣Translation =𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑣𝑣𝑣𝑣 = der IIIohne 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 − 4 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = −4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −∙ 4 2 𝑇𝑇𝑇𝑇 2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2∙𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠2−Time-optimised ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = −4 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 4 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2 2 2 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣2= 2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 belle 13 Bewegungsgesetze der Translation ohne 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Begrenzungen Bewegungsgesetze der Translation ohne 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑠𝑠𝑠𝑠Begrenzungen = 2 + 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 − 2=𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃− 4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃22∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 =−2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 4 ∙ 𝑡𝑡𝑡𝑡 −4 2 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 2 − 4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑎𝑎𝑎𝑎 = −4𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 2 𝐼𝐼𝐼𝐼 −𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑠𝑠𝑠𝑠∙ Geschwindigkeitsoptimal 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 20 𝑃𝑃𝑃𝑃 2 2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20 12 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡2 ∙−𝑡𝑡𝑡𝑡 2− 2 3 ∙ 𝑡𝑡𝑡𝑡33 ∙ 𝑡𝑡𝑡𝑡 3 𝑣𝑣𝑣𝑣 =𝑣𝑣𝑣𝑣6 𝑡𝑡𝑡𝑡2−∙ 𝑡𝑡𝑡𝑡6− 6 3 ∙ 𝑡𝑡𝑡𝑡32 ∙ 𝑡𝑡𝑡𝑡 2 𝑎𝑎𝑎𝑎 =𝑎𝑎𝑎𝑎6= 6 2 − 𝑎𝑎𝑎𝑎= 2 − 12 3 ∙ 𝑡𝑡𝑡𝑡3 ∙ 𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 20 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑇𝑇𝑇𝑇6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 20 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 3 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 =𝑠𝑠𝑠𝑠6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃20 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 ∙ 𝑡𝑡𝑡𝑡2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎63=∙ 𝑡𝑡𝑡𝑡36 2 −212 − 12 𝑣𝑣𝑣𝑣 =36 𝑠𝑠𝑠𝑠∙𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡22 ∙−𝑡𝑡𝑡𝑡 22− 2 3 ∙ 𝑡𝑡𝑡𝑡33 ∙ 𝑡𝑡𝑡𝑡 3 𝑣𝑣𝑣𝑣 =𝑣𝑣𝑣𝑣6= 6 23 𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡2−∙ 𝑡𝑡𝑡𝑡6− 6 3 ∙ 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠32=∙ 3𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎2 = 𝑡𝑡𝑡𝑡3 2∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡2 − 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣 =𝑇𝑇𝑇𝑇6 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃33∙ 𝑡𝑡𝑡𝑡 2 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡 − 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 ∙ 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑇𝑇𝑇𝑇3 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 = 3 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 3 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 20 2 ∙ 𝑡𝑡𝑡𝑡 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 2 Verlustoptimal 20 2 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ 𝑡𝑡𝑡𝑡 −∙ 6 𝑣𝑣𝑣𝑣 = 6 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠= ∙ 𝑡𝑡𝑡𝑡 − −6 12 2 −𝑠𝑠𝑠𝑠312 ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠3− 2 Loss-optimised 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠= 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 3 ∙ 𝑡𝑡𝑡𝑡 2 3 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 2 3 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 6𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃6 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇− 𝑇𝑇𝑇𝑇∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20 −𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 ∙ 𝑡𝑡𝑡𝑡 33 𝑇𝑇𝑇𝑇∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 3Verlustoptimal 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣6 =𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6 2𝑎𝑎𝑎𝑎∙ 𝑡𝑡𝑡𝑡 2− ∙ 𝑡𝑡𝑡𝑡 − 6 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 6 12 𝑡𝑡𝑡𝑡 6 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 6 − 12 ∙ 𝑡𝑡𝑡𝑡 3 2 3 3 2 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

2𝜋𝜋𝜋𝜋

𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

2𝜋𝜋𝜋𝜋

𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣2𝜋𝜋𝜋𝜋 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋∙∙ �1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 = 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2 𝑠𝑠𝑠𝑠∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 − �1𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠==�𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡� 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 =𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣= ∙ �1 − � 2 𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇= 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 � 2𝜋𝜋𝜋𝜋 = 𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡�� ∙ �1 𝑡𝑡𝑡𝑡� − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 Sinus Sine 13 Bewegungsgesetze ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 2𝜋𝜋𝜋𝜋 2 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �Sinus 𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 =Translation �1ohne − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡� 𝑎𝑎𝑎𝑎 =𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋 � 𝑠𝑠𝑠𝑠Tabelle 𝑡𝑡𝑡𝑡��function 𝑣𝑣𝑣𝑣 =der ∙ �1∙− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �Begrenzungen 2 𝑠𝑠𝑠𝑠 Tabelle 13 Bewegungsgesetze der Translation ohne Begrenzungen 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃13 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ohne 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Tabelle Bewegungsgesetze der Translation Begrenzungen 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋13 Motion laws of translation 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Table without limitations ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡� 𝑎𝑎𝑎𝑎 = 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠− 𝑠𝑠𝑠𝑠 �𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡� 𝑎𝑎𝑎𝑎 = 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 2 2𝜋𝜋𝜋𝜋 � 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡� ∙ �1 − �𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋= 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 2 2 𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

20 20 20

𝑎𝑎𝑎𝑎

tion, Vorgabewerte: sPos , amax

1 1𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 3 2 − ∙ 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡 −∙ ∙ 𝑇𝑇𝑇𝑇 3 1 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑡𝑡𝑡𝑡∙ − ∙∙ 𝑡𝑡𝑡𝑡Verlustoptimal ∙ 𝑡𝑡𝑡𝑡 3 3 𝑇𝑇𝑇𝑇3Loss-optimised 𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 3 ∙ ∙ 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 �𝑡𝑡𝑡𝑡∙ �𝑡𝑡𝑡𝑡 − − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡��𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

ment ment ment

SegSegSegSegment SegSegment Segment ment

SegSegSegment ment ment

Seg-SegSegmentment ment

Segment Seg-

ment

Segment

Diese Bewegungsgesetze kommen bei translatorischenDescription Bewegungen zur load Anwendung, wenn d of the cycle dung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die P , Vorgabewerte: sPos , amax slation, Vorgabewerte: ssPoss,Pos amax Beschleunigung die zulässige Beschleunigung nicht überschreitet. Translation, Vorgabewerte: amax max rgabewerte: smotion , amax nslation, Vorgabewerte: Pos anslation, Vorgabewerte: sPos a,max Pos nslation, Vorgabewerte: , ,a,,aa Pos max These lawssare translatory movements if the desired Bewegungen position leadszur to Anwendung, an excee- wenn die ge Diese Bewegungsgesetze kommen bei translatorischen Translation, Vorgabewerte: sPos Translation, Vorgabewerte: sused ainmax Pos , max g Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positi Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die g Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung ding of the permissible acceleration within the required positioning time. Then the positioning time ungsgesetz Parameter Weg Geschwind Translation, Vorgabewerte: sDiese , aBewegungsgesetze Pos max Diese Bewegungsgesetze kommen bei kommen translatorischen BewegungenBewegungen zur Anwendung, wenn die gewü Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, translatorischen zur Anwendung, wew Anwendung Positionierzeit zueiner einer Überschreitung der zulässigen Beschleunigung führt. Dann wird Diese Bewegungsgesetze kommen beibei translatorischen Bewegungen zur Anwendung, wenndie diePosi gew Beschleunigung die zulässige Beschleunigung nicht überschreitet. endung Positionierzeit zu Überschreitung der zulässigen Beschleunigung führt. Dann Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, we Positionierzeit zuBeschleunigung einer Überschreitung derBeschleunigung zulässigen Beschleunigung führt.führt. Dann wird die Position wendung is extended so that thePositionierzeit maximum acceleration does not exceed the permissible acceleration. Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wiw nwendung Positionierzeit zu einer Überschreitung Beschleunigung führt. Dann wir wendung zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die zulässige nicht überschreitet. Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung Dann wird die Positio Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wennwird die ge Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann Beschleunigung die zulässige Beschleunigung nicht überschreitet. Beschleunigung die zulässige Beschleunigung nicht überschreitet. Beschleunigung die zulässige Beschleunigung nicht überschreitet. Beschleunigung die zulässige Beschleunigung Beschleunigung Beschleunigung die die zulässige zulässige Beschleunigung Beschleunigung nicht nicht überschreitet. überschreitet. Anwendung Überschreitung der zulässigen Beschleunigung wird die Posit Translation, Vorgabewerte: sPos ,Positionierzeit a 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 zu einer max Beschleunigung die zulässige Beschleunigung nicht überschreitet.führt. Dann sgesetz Parameter Weg Geschwindigke 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 =� Bewegungsgesetz Parameter Weg Geschwindigk Beschleunigung zulässige Beschleunigung nicht überschreitet. Translation, Vorgabewerte: ssPos aa𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 max Translation, settings: S,Pos , αmax 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 die Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der Translation, Vorgabewerte: , PosDiese max 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Parameter Weg Geschwindigke egungsgesetz Weg Ge wegungsgesetz Parameter Weg Gesch etzBewegungsgesetz Parameter Parameter Weg Geschwindigkeit Anwendung ewegungsgesetz Parameter Weg Gesc Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung Dann die Positionierzeit so verlängert, dass die wegungsgesetz Parameter Weg Gesc ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 wird I bei translatorischen 𝑠𝑠𝑠𝑠 = zurführt. 𝑣𝑣𝑣𝑣 Bewegungsgesetz Parameter Weg Geschw Diese Bewegungsgesetze kommen Bewegungen Anwendung, wenn die gewünschte Position innerhalb SegBeschleunigung die zulässige Beschleunigung nicht überschreitet. Bewegungsgesetz Parameter Weg Geschwindigk Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur2Anwendung, wenn die gewünschte Position innerhalbder de Motion law Parameter Distance 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positionierzeit so verlängert, dass die 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ment Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positionierzeit so verlängert, dass die =� 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ==�𝑇𝑇𝑇𝑇� 𝐼𝐼𝐼𝐼 = 4 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼die Beschleunigung nicht 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Beschleunigung Beschleunigung diezulässige zulässige Beschleunigung nicht überschreitet. überschreitet. Bewegungsgesetz Parameter Weg Beschleun 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit = � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑠𝑠𝑠𝑠 = 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = � 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 I 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣==𝑎𝑎𝑎𝑎 � = 𝑇𝑇𝑇𝑇 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz Weg Geschwindigkeit Beschleun 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = �Parameter 2 � 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 � 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 = 2 � 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 = Bewegungsgesetz Parameter Weg Geschwindigkeit Beschleu 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑠𝑠𝑠𝑠 2= 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃4 � 𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = � 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 = 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙− 𝑠𝑠𝑠𝑠2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = =𝑎𝑎𝑎𝑎∙∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 222 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III II II𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠∙ = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =� 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =4𝐼𝐼𝐼𝐼� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣==� 𝑠𝑠𝑠𝑠=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 I𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡∙ 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 windigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 ∙𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼= 𝑎𝑎𝑎𝑎∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡 I 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2== 𝑠𝑠𝑠𝑠4 𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∙�𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠 2 = 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 222 𝐼𝐼𝐼𝐼 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 2 =� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 44 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠 = 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �4 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 22 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �4 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = == 4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � II 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �� 𝑎𝑎𝑎𝑎4 III 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙2∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑠𝑠𝑠𝑠� ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 � 𝑎𝑎𝑎𝑎 ∙+ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃22 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝑣𝑣𝑣𝑣 = ∙𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 = III 𝑠𝑠𝑠𝑠 = + 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 �4 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣= =� �∙ 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal III 𝑠𝑠𝑠𝑠 = + 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑇𝑇𝑇𝑇�𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � �4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 igkeitsoptimal 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − =� 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 =III 2 + �𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 2 = �𝑠𝑠𝑠𝑠 ∙𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑎𝑎𝑎𝑎 III = �𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 + 𝑠𝑠𝑠𝑠 ∙ − 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑎𝑎𝑎𝑎� 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎= � 2 � 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑠𝑠𝑠𝑠 = + 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 2 2 � � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 chwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 ∙𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎= Time-optimised 2 ∙ ∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 III+ = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −∙ 𝑎𝑎𝑎𝑎− ∙ 𝑡𝑡𝑡𝑡= III + ∙∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑃𝑃𝑃𝑃= ∙𝑎𝑎𝑎𝑎∙𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎� = � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 == 𝑠𝑠𝑠𝑠III ∙2𝑎𝑎𝑎𝑎∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡� − ∙∙ �𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 + −+ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = Geschwindigkeitsoptimal 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = = ∙∙𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎= + ∙∙𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 eschwindigkeitsoptimal 2 ∙ ∙𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑠𝑠𝑠𝑠 +𝑎𝑎𝑎𝑎∙� 𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −22−−2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � � =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠� ∙∙𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 itsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 schwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +�𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 hwindigkeitsoptimal 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎III = ∙− 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠� 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2 1 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22 2 3 22 2 2 22 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 == �6�6 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡 − der 𝑠𝑠𝑠𝑠 =innerhalb ∙ geforderten ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 orischen Bewegungen Anwendung, wenn die gewünschte Position innerhalb 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 schen Bewegungen zurzur Anwendung, die gewünschte Position 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 der 1geforderten 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 wenn 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃6𝑠𝑠𝑠𝑠 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 1= ∙ 𝑡𝑡𝑡𝑡 − ∙ maximale ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑎𝑎𝑎𝑎 �6 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ässigen Beschleunigung führt. Dann wird die Positionierzeit so verlängert, dass die 𝑎𝑎𝑎𝑎 1 𝑎𝑎𝑎𝑎 = 2 �6 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 sigen Beschleunigung führt. Dann𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 wird die dass 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 die maximale 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6 6 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎Positionierzeit ∙ 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠2 − 𝑠𝑠𝑠𝑠so = verlängert, 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =3 ∙ 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 − ∙ ∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 �6𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 �𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 == ∙3𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 � 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎3 1 𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 nicht überschreitet. 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 11 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 �6 = 𝑇𝑇𝑇𝑇 � ht überschreitet. 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1 = 66𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 3 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎2 3=� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 ∙ 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣3= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 2 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 22 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑠𝑠 = − ∙ ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 3 𝑎𝑎𝑎𝑎 = ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 ∙∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 − ∙ ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 2 ∙ 𝑡𝑡𝑡𝑡 −3 ∙ 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 optimal 8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 1 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 3 2 innerhalb 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 innerhalb 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 hen Bewegungen zur Anwendung, wenn diedie Position der 2geforderten 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 383� 𝑠𝑠𝑠𝑠gewünschte ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎3𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1∙1 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal chen Bewegungen zur Anwendung, wenn gewünschte Position der Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡geforderten ∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡33 3 2− � 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 8 3∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠= 1 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 23 = −− ∙ 𝑇𝑇𝑇𝑇∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣wird =� 𝑠𝑠𝑠𝑠� 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 3311 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑎𝑎𝑎𝑎 Geschwindigkeit Beschleunigung 88Positionierzeit Verlustoptimal 𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 3 3 𝑣𝑣𝑣𝑣 = Geschwindigkeit Beschleunigung en Beschleunigung führt. Dann wird die so verlängert, dass die maximale igen Beschleunigung führt. Dann die Positionierzeit so verlängert, dass die maximale 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = − ∙ ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = − ∙ Verlustoptimal ischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃die ∙ 𝑎𝑎𝑎𝑎gewünschte ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = − ∙ ∙ 𝑡𝑡𝑡𝑡 8 Bewegungen zur Anwendung, wenn Position innerhalb der geforderten Verlustoptimal Loss-optimised 2 𝑇𝑇𝑇𝑇 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 33 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal überschreitet. 3 ∙ 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 die maximale ht überschreitet. 𝑠𝑠𝑠𝑠� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =� sigen Beschleunigung führt.3wird Dann wird die Positionierzeit so verlängert, dass �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣8 = Beschleunigung führt. Dann die so verlängert, dass die maximale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 mal Bewegungen zurAnwendung, Anwendung, die gewünschte innerhalb der geforderten 𝑎𝑎𝑎𝑎� ewegungen zur wenn die gewünschte innerhalb der geforderten 𝑣𝑣𝑣𝑣 =8 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙Position 𝑎𝑎𝑎𝑎Position 3𝑠𝑠𝑠𝑠3 𝑇𝑇𝑇𝑇Positionierzeit = 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇wenn 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣= � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � cht überschreitet. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 die maximale 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 dass 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ustoptimal 𝑇𝑇𝑇𝑇Positionierzeit = �2𝜋𝜋𝜋𝜋 erschreitet. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung führt. Dann wird die Positionierzeit so verlängert, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 8 𝑇𝑇𝑇𝑇 = schleunigung führt. Dann wird die so verlängert, dass die maximale 𝑇𝑇𝑇𝑇 = 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 Verlustoptimal �𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 8Geschwindigkeit 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 erlustoptimal 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 88�𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 lustoptimal 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 rschreitet. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 2𝜋𝜋𝜋𝜋 chreitet. ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡�� = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 stoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 Vorgabewerte: sPos , amax𝑠𝑠𝑠𝑠� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙��𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 � = 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Translation, 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � = 𝑡𝑡𝑡𝑡�� = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� = 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇�𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − Translation, Vorgabewerte: s , a Pos max Geschwindigkeit Beschleunigung 2 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 𝜋𝜋𝜋𝜋 Sinus Translation, Vorgabewerte: s , a 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Beschleunigung 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇Geschwindigkeit 2𝜋𝜋𝜋𝜋max 2 𝑣𝑣𝑣𝑣2 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �Diese ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 = 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 = 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =Pos 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃de 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 bei translatorischen Bewegungen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetze zur Anwendung, wenn die gewünschte Position innerhalb 2𝑠𝑠𝑠𝑠 � ∙ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇 = 2 2 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠einer ∙ kommen �= 𝑠𝑠𝑠𝑠Beschleunigung = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 zur −führt. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �wird 𝑣𝑣𝑣𝑣verlängert, = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 bei translatorischen Diese Bewegungsgesetze kommen Bewegungen Anwendung, wenn die gewünschteso Position innerhalb de∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 � Geschwindigkeit 𝑇𝑇𝑇𝑇 = 2𝜋𝜋𝜋𝜋 Anwendung Positionierzeit derBeschleunigung zulässigen Beschleunigung die𝑡𝑡𝑡𝑡�� Positionierzeit dass � 2 Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese Bewegungsgesetze kommen bei Bewegungen Anwendung, die gewünschte Position innerhalb de 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetze Translation dertranslatorischen Beschleunigung Sinus 14 𝜋𝜋𝜋𝜋mit SinusTabelle 𝑎𝑎𝑎𝑎Begrenzung 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 zur 2𝜋𝜋𝜋𝜋 Dann 2𝜋𝜋𝜋𝜋 die 𝑇𝑇𝑇𝑇wenn 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎Überschreitung � Sine function 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 der = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙zu 𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 � Anwendung 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 Positionierzeit zu einer Überschreitung der zulässigen zulässigen Beschleunigung führt. Dann Dann wird die Positionierzeit Positionierzeit so so verlängert, verlängert,𝑇𝑇𝑇𝑇dass dass die 𝑇𝑇𝑇𝑇 = 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung die zulässige Beschleunigung nicht überschreitet. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Anwendung 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Positionierzeit zu einer Überschreitung der Beschleunigung führt. wird die 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋derBeschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 2𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 2 � 𝜋𝜋𝜋𝜋 Sinus Tabelle 14 Bewegungsgesetze 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � ∙� 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 die zulässige zulässige Beschleunigung nicht überschreitet. überschreitet. 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 Translation mit Begrenzung der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚nicht 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung die Beschleunigung 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎 �2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung Beschleunigung =� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣2𝜋𝜋𝜋𝜋 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 14 der Translation mit Begrenzung der Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 law Velocity Acceleration 2 Bewegungsgesetze 𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋 = Parameter Weg Geschwindigkeit Beschleu ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 2Motion = = Tabelle 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡Bewegungsgesetz 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼Weg 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃der 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 21 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 Bewegungsgesetz Parameter Geschwindigkeit Beschleu 2𝜋𝜋𝜋𝜋 2Parameter 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 der2Translation 2𝑎𝑎𝑎𝑎2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚14Bewegungsgesetz Tabelle Bewegungsgesetze der Translation mit Begrenzung Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Weg Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇�𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣∙ = 14 Bewegungsgesetze Begrenzung der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎∙�= 𝑠𝑠𝑠𝑠−=∙ 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleu − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙Beschleunigung = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 �1 = 𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙� 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎2 ∙∙14 ∙𝐼𝐼𝐼𝐼 2Bewegungsgesetze 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= � der 𝑎𝑎𝑎𝑎�𝑡𝑡𝑡𝑡 = −𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣Translation = 𝑠𝑠𝑠𝑠mit 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡mit 𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎 � ∙ ∙𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −− ∙𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 −𝑎𝑎𝑎𝑎 = ∙=𝑎𝑎𝑎𝑎∙∙∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 21 Tabelle Begrenzung der Beschleunigung 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = � 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 2 21 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 � 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 2𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 us 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 𝑇𝑇𝑇𝑇= = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎==𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 � 𝜋𝜋𝜋𝜋 ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣 =𝑇𝑇𝑇𝑇= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ∙ 𝑎𝑎𝑎𝑎I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = = � 21 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �2𝜋𝜋𝜋𝜋 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡 22 nus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 21 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ewegungsgesetze Begrenzung der Beschleunigung I 𝑠𝑠𝑠𝑠 = 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 us =mit ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚der2Translation =� �4𝜋𝜋𝜋𝜋𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃mit 2 Beschleunigung 2 der lle 14 Bewegungsgesetze Translation Begrenzung der 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = −𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung 21 𝜋𝜋𝜋𝜋 s𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 21 = −𝑎𝑎𝑎𝑎 � 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇= 𝑠𝑠𝑠𝑠�4 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 belle 14 Bewegungsgesetze der𝑣𝑣𝑣𝑣Translation mit−Begrenzung der Beschleunigung 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2−𝑎𝑎𝑎𝑎 ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎 = Time-optimised = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 gungsgesetze der Translation mit Begrenzung der Beschleunigung III 𝑠𝑠𝑠𝑠 = + ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = −𝑎𝑎𝑎𝑎 elle 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � Geschwindigkeitsoptimal 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎�𝑠𝑠𝑠𝑠 1𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 + �𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡Beschleunigung − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = = �𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 mit = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 3 32𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 1Bewegungsgesetze 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = e2∙∙14 der 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠Translation Begrenzung = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡− 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 =−𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎 ∙−III 𝑡𝑡𝑡𝑡 ∙− 𝑎𝑎𝑎𝑎−𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= ∙ 2𝑡𝑡𝑡𝑡2 2∙ +𝑡𝑡𝑡𝑡 2�𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎der 𝑡𝑡𝑡𝑡− 𝑡𝑡𝑡𝑡∙2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎= 𝑣𝑣𝑣𝑣∙ 𝑎𝑎𝑎𝑎∙= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼21 = 𝑎𝑎𝑎𝑎22𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣==�� 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ −∙ Geschwindigkeitsoptimal ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 3𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 21 21 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = = �6 �6 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 1 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 21 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑠𝑠𝑠𝑠 =2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 2 − ∙21 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 2 ∙ 𝑡𝑡𝑡𝑡22 − 11 3 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑡𝑡𝑡𝑡∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡22 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 == 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡== 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙= −− 2 221 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡∙ − 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣∙= ∙ 𝑡𝑡𝑡𝑡 − 3 ∙∙𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡3𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 38 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 −∙ 𝑡𝑡𝑡𝑡 2 ∙ 𝑡𝑡𝑡𝑡 2 𝑎𝑎𝑎𝑎 = − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 �= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎= =𝑣𝑣𝑣𝑣� 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣= 𝑡𝑡𝑡𝑡 ∙∙− 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 − 2∙ 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 88 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡−− 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚−−2 2 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣==𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎==𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = = �2𝜋𝜋𝜋𝜋 �2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡� 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1∙ �1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� −− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡�� ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡 − − 2𝜋𝜋𝜋𝜋 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 =� 𝑡𝑡𝑡𝑡�� = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎�𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ��= 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣�= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �

𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus �2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝜋𝜋𝜋𝜋 Sinus Sine function 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Tabelle2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung − �𝑠𝑠𝑠𝑠 � 14 ∙ �1 −− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � der 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎Translation 𝑡𝑡𝑡𝑡��𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 == 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙ 𝑠𝑠𝑠𝑠 Tabelle 14𝑡𝑡𝑡𝑡�� Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung 𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙Begrenzung �1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � Beschleunigung 𝑡𝑡𝑡𝑡�� = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 mit 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle Bewegungsgesetze der

2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Table 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇limitation 2𝜋𝜋𝜋𝜋 14 Motion laws of translation of 2𝜋𝜋𝜋𝜋 acceleration 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 with 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡�� ∙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = ∙𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡� � 𝑡𝑡𝑡𝑡� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 − � 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 � hleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 unigung 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 21 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 21 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎 =21𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

nigung unigung

𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

ment

Segment Seg-Seg- SegSeg- mentment ment ment

Segment Segment SegSegSegSeg- SegSegSeg- ment ment ment Segment ment ment ment

tion, Vorgabewerte: sPos , amax , vmax Description load , Vorgabewerte: sPosof, the amax , vcycle max Translation, Vorgabewerte: s , amax , ,v,vvmax Diese kommen bei translatorischen Bewegungen zur Anwendung, Pos Bewegungsgesetze Translation, Vorgabewerte: s Translation, Vorgabewerte: sPos Pos, ,aamax max max max DiesePositionierzeit Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, di zu einer Überschreitung der zulässigen Beschleunigung und derwenn zuläss These motion laws are used in translatory movements if the desired position leads to an exceeding dung Positionierzeit zuBewegungsgesetze einer Überschreitung der zulässigen Beschleunigung und der zulässigen Gew Diese kommen bei translatorischen Bewegungen zur Anwendung, Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit so verlä Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, Diese Bewegungsgesetze translatorischen Bewegungen zur Anwendung, w ganslation,ofVorgabewerte: sPosPositionierzeit , asPos ,eine vmax maxand the permissible acceleration permissible velocity eingefügt within the required positioning time.so Averlängert, der Beschleunigung und der zulässi und dienicht Positionierzeit Translation, ,max a vzu max max ,lation, Vorgabewerte: sPosVorgabewerte: ,Bewegungsablaufs a Vorgabewerte: sPos , ,avmax ,Positionierzeit v,amax max max Translation, Vorgabewerte: sPos a,max ,vv,the Translation, Vorgabewerte: sPositionierzeit a v,Konstantfahrtphase Pos max max Pos max und die Geschwindigkeit dieÜberschreitung zulässige Geschwindigkeit überschreiten. Translation, Vorgabewerte: s , , zu einer Überschreitung zulässigen Beschleunigung undder derzulässig zulässd max zueiner einer Überschreitung der zulässigen zulässigen Beschleunigung und Anwendung Anwendung Anwendung Bewegungsablaufs eine eingefügt und die Positionierzeit so verlän und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. constant velocity segment is then inserted in theKonstantfahrtphase middle of the movement sequence andPositionierzeit the positio- so Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit soverlän verlä Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendun Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, we und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die Diese Bewegungsgesetze kommen bei translatorischen Bewegungen we Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gew Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, ww Translation, Vorgabewerte: s , a und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. und die Geschwindigkeit Geschwindigkeit nicht überschreiten. Pos max , vmax ning time is extended so that the acceleration does not exceed the permissible acceleration and the Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulä Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässige ungsgesetz Parameter Weg Ge Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässigen Gesch Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässig Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässig Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässigen Geschwi Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung nwendung gesetz Parameter Weg Geschwin Anwendung Anwendung gndung Bewegungsablaufs eine Konstantfahrtphase eingefügt und diedie Positionierzeit so verläng Anwendung velocity does not Bewegungsablaufs exceed the permissible velocity. Bewegungsablaufs eine Konstantfahrtphase eingefügt und Positionierzeit so ve kommen beieingefügt translatorischen Bewegungen zur Anwendung, wenn die ge eine Konstantfahrtphase eingefügt und die eine Konstantfahrtphase eingefügt und diePositionierzeit Positionierzeit verlän eine Konstantfahrtphase und die Positionierzeit so verlängert, das Translation, Vorgabewerte: sPos ,Bewegungsablaufs aBewegungsablaufs ,eine vmaxBewegungsgesetze maxDiese Konstantfahrtphase eingefügt und die Positionierzeit so verlängert, dass d Bewegungsablaufs eine Konstantfahrtphase eingefügt und die sosoverläng Bewegungsgesetz Parameter Weg Ges Bewegungsgesetz Bewegungsablaufs Parameter Ges und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. Bewegungsgesetz Parameter Weg Ge Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung und der zulässigen Geschw und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. die Geschwindigkeit die zulässige Bewegungen Geschwindigkeit nicht und die Geschwindigkeit die zulässige Geschwindigkeit nichtzurüberschreiten. Anwendung 𝑣𝑣𝑣𝑣 Translation, Vorgabewerte: sPosund , amax , vmax 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Diese Bewegungsgesetze kommen bei translatorischen Anwendung, wenn die gewünschte Position innerhalb 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsablaufs eingefügt und die∙ 𝑡𝑡𝑡𝑡Positionierzeit so verlängert, dass dd 2 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 =SPositionierzeit 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 = zu, Veiner Überschreitung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = Ieine Konstantfahrtphase 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼,𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Geschwindigkeit führt. Dann wird in der zulässigen Beschleunigung und der 22 zulässigen Translation, settings α Anwendung Pos max 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼sPos =Bewegungsablaufs und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣d Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣eine Bewegungsgesetz Parameter Weg Translation, Vorgabewerte: ,𝑎𝑎𝑎𝑎amax ,max v𝑎𝑎𝑎𝑎max 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Konstantfahrtphase eingefügt und die Positionierzeit so verlängert, dass die Beschleunigung dieGesc zulässi 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠und 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 Bewegungsgesetz WegBeschleunigung Ges 𝑇𝑇𝑇𝑇Parameter Bewegungsgesetz Parameter Weg Gesc ewegungsgesetz Parameter Weg Positionierzeit zu=einer Überschreitung der zulässigen führt. Dann wird in G d gungsgesetz Parameter Weg Geschwind IIGeschwindigkeit Bewegungsgesetz Weg Gesc 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼= =𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= gesetz Weg Geschwindigk 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= = ∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2Geschwindigkeit und die Geschwindigkeit die zulässige nicht überschreiten. 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼 = Anwendung Parameter Parameter =der zulässigen Seg-

𝑚𝑚𝑚𝑚

Segment

Bewegungsgesetz Anwendung

Segment

𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼wenn 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsablaufs Konstantfahrtphase eingefügt undBewegungen die Positionierzeit verlängert, dassdie diegewünschte Beschleunigung die zulässi 22 Diese Bewegungsgesetze kommen bei translatorischen zur Anwendung, Position innerh 2 so2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠Parameter 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎eine Parameter Weg Distance Geschwindig 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ment und Geschwindigkeit die Geschwindigkeit nicht 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Positionierzeit Überschreitung der zulässigen Beschleunigung und der zulässigen Geschwindigkeit führt. Dann wir 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚zulässige 𝑣𝑣𝑣𝑣die 𝑎𝑎𝑎𝑎+ 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑇𝑇𝑇𝑇 =zu𝑇𝑇𝑇𝑇 einer 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II Weg Geschwindigkeit Beschleun = 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 =Parameter 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 = 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = − I eingefügt 𝑣𝑣𝑣𝑣 die 𝑎𝑎𝑎𝑎 Positionierzeit so dass die Beschleunigung die zu 𝑣𝑣𝑣𝑣eine 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣Bewegungsablaufs 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 II 𝑠𝑠𝑠𝑠 = und + 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2∙𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼verlängert, 𝐼𝐼𝐼𝐼 2𝐼𝐼𝐼𝐼 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎Konstantfahrtphase 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣Parameter 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 2𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇= = 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼und = 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 ∙2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz WegIIIIIII IGeschwindigkeit2nicht Geschwindigkeit Beschleuni 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼− 𝐼𝐼𝐼𝐼−= 𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 die Geschwindigkeit zulässige überschreiten. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙= 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼= 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎= 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎∙𝑠𝑠𝑠𝑠 Idie 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼= = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 = 𝐼𝐼𝐼𝐼= 𝑚𝑚𝑚𝑚 = ∙∙𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 22+ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 III I = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚∙𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙𝑠𝑠𝑠𝑠2𝑎𝑎𝑎𝑎= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼= 𝑣𝑣𝑣𝑣 = I𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎= 𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2+ 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 2 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 2 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 = 2 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 I 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎 2 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz Parameter Weg Geschwindigkeit Beschl 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 22 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 +𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 ∙=𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑠𝑠𝑠𝑠 = 2 𝑠𝑠𝑠𝑠 2 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 =𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 I 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼+ 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 2 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 =𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣2𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 22+ 2 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 ∙∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 − 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣−𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇=𝐼𝐼𝐼𝐼− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠= = +𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝐼𝐼𝐼𝐼=𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 + 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚+ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 IIIIII𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −= = + 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼= 𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚 indigkeitsoptimal 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠II= − 𝑠𝑠𝑠𝑠 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 ∙= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙∙ 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2− 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼222 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣=𝑣𝑣𝑣𝑣=𝑣𝑣𝑣𝑣=𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 −− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 ∙+ 𝑡𝑡𝑡𝑡− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 =−𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 −𝑚𝑚𝑚𝑚 IIII 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡∙𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼= = + 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣∙+ 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎IIII 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = +III + 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣= 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃 − 𝑎𝑎𝑎𝑎 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 + 𝑣𝑣𝑣𝑣 − III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑎𝑎𝑎𝑎 2 ∙ 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − 𝑡𝑡𝑡𝑡 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 gkeitsoptimal 2 ∙ 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 2 ∙ 𝑎𝑎𝑎𝑎 2 II 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣 ∙ 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 + 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 2 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼2 III 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 − 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 =𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 Geschwindigkeitsoptimal ∙ 𝑡𝑡𝑡𝑡 − 𝑣𝑣𝑣𝑣 = ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22∙∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 2=∙ 𝑎𝑎𝑎𝑎+𝑠𝑠𝑠𝑠2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙2𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎+ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II I𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = Geschwindigkeitsoptimal 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼2 ∙ 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 wegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten 𝑣𝑣𝑣𝑣 III− Position =der 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡innerhalb − 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22 der + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 egungenGeschwindigkeitsoptimal zur Anwendung, die gewünschte latorischen Bewegungen zurwenn Anwendung, wenn gewünschte geforderten = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 die 𝑣𝑣𝑣𝑣 ∙=𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃innerhalb +𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 −geforderten 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣Position 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Time-optimised 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑣𝑣𝑣𝑣22𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣= ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇=𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ++ 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +𝑣𝑣𝑣𝑣 = 𝑇𝑇𝑇𝑇 2des 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎Mitte 2=𝐼𝐼𝐼𝐼 2𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III III −Position II − 𝑣𝑣𝑣𝑣2∙des 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = − + 𝑣𝑣𝑣𝑣− ∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼22𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣==𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = +𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚− ∙Mitte 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃+ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑣𝑣𝑣𝑣∙𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼Mitte 𝐼𝐼𝐼𝐼 − chleunigung undder derzulässigen zulässigen Geschwindigkeit führt. Dann wird in 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼∙− 𝐼𝐼𝐼𝐼− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 orischen Bewegungen zurAnwendung, Anwendung, wenn die gewünschte innerhalb der geforderten zulässigen Beschleunigung und der zulässigen führt. Dann wird in des hleunigung und Dann wird in der III 𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 = +Geschwindigkeit 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡der 𝑡𝑡𝑡𝑡− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 + 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 orischen Bewegungen zur die innerhalb geforderten 𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎führt. 𝑣𝑣𝑣𝑣∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡=𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 ∙∙∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 III III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙= 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃=Position +𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙der 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼− ∙ 𝑡𝑡𝑡𝑡der 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =wenn 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼−𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼2 Geschwindigkeitsoptimal ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚gewünschte 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼2 𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 +𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 − Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III+ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠III hwindigkeitsoptimal 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚− 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑠𝑠𝑠𝑠+ ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙2𝑎𝑎𝑎𝑎∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 2=∙ 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚∙2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣der 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 gkeitsoptimal lässigen Beschleunigung und zulässigen Geschwindigkeit führt. Dann wird in der Mitte des 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 eeschwindigkeitsoptimal eingefügt und die Positionierzeit so verlängert, dass die Beschleunigung die zulässige Beschleunigung 2 ∙ 𝑎𝑎𝑎𝑎 2 die Positionierzeit soverlängert, verlängert, dass die Beschleunigung die zulässige Beschleunigung 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 die Positionierzeit so dass die Beschleunigung die zulässige Beschleunigung 2 ∙ 𝑎𝑎𝑎𝑎 2 ässigen Beschleunigung und der zulässigen Geschwindigkeit führt. Dann wird in der Mitte des 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 der Bewegungsablauf die𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚die maximal zulässige bestimmt, h 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 durch durch 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Wird Wird der Bewegungsablauf die=durch maximal zulässige Geschwindigkeit hat dasha v Wird Bewegungsablauf durch maximal zulässige Geschwindigkeit bestimmt, 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃der =der + der Bewegungsablauf die maximal Geschwindigkeit bestimmt, ha = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚bestimmt, IIIdurch 𝑠𝑠𝑠𝑠durch + 𝑣𝑣𝑣𝑣die ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼bestimmt, 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − zulässige 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡zulässige 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − zulässige Wird Bewegungsablauf die𝑠𝑠𝑠𝑠 maximal Geschwindigkeit hat das𝑣𝑣𝑣𝑣 verlustoptimale Bewegungsgesetz Wird der maximal zulässige Geschwindigkeit ngefügt undnicht diePositionierzeit Positionierzeit soWird verlängert, dass die Beschleunigung hwindigkeit überschreiten. 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Bewegungsablauf 𝑎𝑎𝑎𝑎dass ht überschreiten. Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎die 2 bestimmt, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ngefügt und die so verlängert, die Beschleunigung die zulässige Beschleunigung t überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte indie dhk If the motion sequence is determined by theDaher maximum permissible en bei translatorischen Bewegungen zur Anwendung, wenn diegeschwindigkeitsoptimalen gewünschte Position innerhalb der mehrmehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher insollte diesem Fa mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. solltegeforderten in diesem Fall auf sollte das geschwindigkeitsoptim mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher in mehr gegenüber dem Bewegungsgesetz. Daher sollte in die Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, hat mehr gegenüber dem geschwindigkeitsoptimalen Daher sollte dik Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit Bewegungsgesetz. hat das verlustoptimale Bewegungsgesetz ndigkeit nicht überschreiten. ndigkeit nicht überschreiten. Bewegungsgesetz zurück gegriffen werden. velocity, the loss-optimised motion law noinbestimmt, advantage over bestimmt, the Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit hat das in reitung der zulässigen Beschleunigung und der zulässigen Geschwindigkeit führt. Dannhas wird der Mitte des Bewegungsgesetz zurück gegriffen werden. Bewegungsgesetz zurück gegriffen werden. Bewegungsgesetz zurück gegriffen werden. mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptima mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in verlu diese Bewegungsgesetz zurück gegriffen werden. Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, ha Bewegungsgesetz zurück gegriffen werden. Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, hat mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher solltelaw in diesem Fall au Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, hat das ver time-optimised motion law. In this case, the time-optimised motion tfahrtphase eingefügt und die Positionierzeit so verlängert, dass die Beschleunigung die zulässige Beschleunigung Bewegungsgesetz zurück gegriffen werden. Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, hat Geschwindigkeit Beschleunigung Geschwindigkeit Beschleunigung Bewegungsgesetz zurück gegriffen werden. Geschwindigkeit Verlustoptimal Wird der Bewegungsablauf durch dieBeschleunigung maximal zulässige Geschwindigkeit bestimmt, hatbestimmt, das verlustoptimale Bewegungsges mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte inFall die Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit hat das verlus Bewegungsgesetz zurück gegriffen werden. mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in dies Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimm should be used. ässige Geschwindigkeit nicht überschreiten. mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher in diesem mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in dies Geschwindigkeit Beschleunigung mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in𝑣𝑣𝑣𝑣diesem Fallsollte auf das geschwindigkeitsop 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Loss-optimised Verlustoptimal Verlustoptimal optimal zurück gegriffen werden. mehr gegenüber Bewegungsgesetz. Daher sollte diesem Fall 𝑇𝑇𝑇𝑇Bewegungsgesetz = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =geschwindigkeitsoptimalen mal Bewegungsgesetz gegriffen 𝑠𝑠𝑠𝑠 werden. = ∙ �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 −werden. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� Bewegungsgesetz. 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �in Daher 𝑡𝑡𝑡𝑡 �� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎au Izurück Verlustoptimal 𝐼𝐼𝐼𝐼 dem 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz zurück gegriffen werden. Bewegungsgesetz gegriffen mehr gegenüber dem geschwindigkeitsoptimalen sollte in Verlustoptimal 2 zurück 𝑎𝑎𝑎𝑎 2 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Bewegungsgesetz zurück gegriffen werden. 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 zurück Bewegungsgesetz werden. 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 =zurück = 𝑣𝑣𝑣𝑣 2 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 = ∙𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚gegriffen 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 2𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 Weg Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz gegriffen werden. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠 �𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 =𝐼𝐼𝐼𝐼 = −𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇 = 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 II 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �� 2 I 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 Verlustoptimal 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 zur Anwendung, die gewünschte innerhalb geforderten == 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =2∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡= ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡 −𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2= 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙der �𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡�𝐼𝐼𝐼𝐼�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣= 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 I 4𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑠𝑠𝑠𝑠2𝑎𝑎𝑎𝑎 �𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = ∙𝑣𝑣𝑣𝑣= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚I Position II 𝑠𝑠𝑠𝑠 = 𝐼𝐼𝐼𝐼= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Bewegungen 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇= 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =2𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡n 𝑣𝑣𝑣𝑣wenn 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎=𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙𝐼𝐼𝐼𝐼− − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣�� Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 2 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋∙∙= 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝐼𝐼𝐼𝐼 ��𝑣𝑣𝑣𝑣 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 =𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠2=2 ∙ �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝐼𝐼𝐼𝐼�𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 = −= 2𝑎𝑎𝑎𝑎 I I𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 stoptimal 𝐼𝐼𝐼𝐼− 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2∙ �1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼a =die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Vorgabewerte: wenn sPos ,2 , v2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 max 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼2 zur und ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 führt. 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��= =− 𝑎𝑎𝑎𝑎 = ² 2 𝑣𝑣𝑣𝑣 + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡2 n Beschleunigung Bewegungen Anwendung, gewünschte innerhalb der geforderten 𝐼𝐼𝐼𝐼 max 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 𝑎𝑎𝑎𝑎II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠== 2 ∙ ∙𝑡𝑡𝑡𝑡Translation, 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙Position 𝑎𝑎𝑎𝑎2𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 nmal der zulässigen Dann der Mitte des 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 =𝑣𝑣𝑣𝑣Geschwindigkeit 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4∙ �𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = − wird 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼�𝐼𝐼𝐼𝐼in 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = ∙𝑣𝑣𝑣𝑣�1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 2 𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠= 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = ∙ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 2 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 wegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑣𝑣𝑣𝑣des 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ewegungen wenn Position der geforderten 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2𝑎𝑎𝑎𝑎 2 zur Anwendung, 22𝑎𝑎𝑎𝑎 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣wird 2 𝜋𝜋𝜋𝜋 2 𝑠𝑠𝑠𝑠die 𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚innerh 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃wenn 𝑣𝑣𝑣𝑣gewünschte erlustoptimal 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 =𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠gewünschte 2 zulässigen 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 n Beschleunigung und führt. Dann in der 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣==𝑣𝑣𝑣𝑣 2 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣∙𝜋𝜋𝜋𝜋zur Iinnerhalb 𝑣𝑣𝑣𝑣= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝜋𝜋𝜋𝜋Geschwindigkeit = 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚− 𝑣𝑣𝑣𝑣Position 𝑣𝑣𝑣𝑣die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋�𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠∙2Bewegungen = ∙𝑣𝑣𝑣𝑣Mitte �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡 �� 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠2geforderten = +=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 ∙�𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡die 𝜋𝜋𝜋𝜋 innerhalb kommen bei∙II Anwendung, wenn Position Iinnerhalb 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼=𝐼𝐼𝐼𝐼Bewegungsgesetze Bewegungen Bewegungen zur zur Anwendung, die die gewünschte der der 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 =Anwendung, ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 = 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡II = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Positionierzeit 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋 tschleunigung und so verlängert, dass die die zulässige 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 gewünschte 𝑇𝑇𝑇𝑇wenn =Geschwindigkeit 𝑇𝑇𝑇𝑇 = 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎II 𝑎𝑎𝑎𝑎Beschleunigung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣wird =² +𝜋𝜋𝜋𝜋𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣in �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 = 𝑣𝑣𝑣𝑣 �1 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 = IPosition 𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 IIIIItranslatorischen − 𝑠𝑠𝑠𝑠geforderten + 𝑣𝑣𝑣𝑣0 − 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣∙= =∙ 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣= ∙�𝐼𝐼𝐼𝐼∙ ∙∙𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠Beschleunigung ∙+ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣führt. 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣II ∙= 22𝑎𝑎𝑎𝑎− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠− = + 𝑣𝑣𝑣𝑣 == 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼+ 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 = 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼∙𝑚𝑚𝑚𝑚 = −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 4𝐼𝐼𝐼𝐼des 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎�+ 2∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼zulässigen 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣2 2𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎− führt. Dann der Mitte 𝑠𝑠𝑠𝑠∙ 4𝑡𝑡𝑡𝑡= = + 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠und = die +und 𝑣𝑣𝑣𝑣und − 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼2𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼+ 𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 eschleunigung der zulässigen Geschwindigkeit führt. Dann wird der Mitte des II𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 Positionierzeit einer der und der zulässigen Geschwindigkeit Dann wir 𝑠𝑠𝑠𝑠die = 2 +𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2Überschreitung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 4II 𝑎𝑎𝑎𝑎zulässigen 2𝑣𝑣𝑣𝑣in 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚der 2 Positionierzeit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋2zu 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 so verlängert, dass die Beschleunigung zulässige Beschleunigung Sinus 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 Anwendung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 4 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 = 0 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 4 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝑣𝑣𝑣𝑣 = ∙ �1 −zu Beschleunigung Beschleunigung und und der der zulässigen zulässigen Geschwindigkeit Geschwindigkeit führt. führt. Dann Dann wird wird in in der der Mitte Mitte des des 𝑣𝑣𝑣𝑣t𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 2 ∙ 𝑎𝑎𝑎𝑎 I = 𝑣𝑣𝑣𝑣 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚und𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝜋𝜋𝜋𝜋 +0 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �� 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣 +𝑠𝑠𝑠𝑠 2 verlängert, 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 t nicht 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼überschreiten. III 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsablaufs eine Konstantfahrtphase eingefügt die Positionierzeit so dass die Beschleunigung die 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋=𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣die 2= 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝜋𝜋𝜋𝜋−= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 = 2zulässige dt nicht die Positionierzeit verlängert, dass 𝑠𝑠𝑠𝑠𝜋𝜋𝜋𝜋2𝑎𝑎𝑎𝑎 =𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣Beschleunigung ∙²𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��2𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Idie = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚00 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 die Positionierzeit so verlängert, dass die zulässige Beschleunigung 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 =nd ++𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 so 2𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠,𝑃𝑃𝑃𝑃und 𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇die 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣− überschreiten. 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼die 𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋 ²= 𝜋𝜋𝜋𝜋die 𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡 ²− 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 II 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 die Geschwindigkeit die zulässige II Geschwindigkeit nicht überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 2 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼v 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 𝑇𝑇𝑇𝑇 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 und nd die Positionierzeit Positionierzeit so so verlängert, verlängert, dass dass Beschleunigung Beschleunigung die zulässige zulässige Beschleunigung Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Translation, Vorgabewerte: s a , 2∙ ∙die 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 Pos max max 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 ² 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 ² 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠+ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = = 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠II ∙ 𝑣𝑣𝑣𝑣2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠2𝑎𝑎𝑎𝑎 + ∙𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜋𝜋𝜋𝜋= −²𝜋𝜋𝜋𝜋− +𝑚𝑚𝑚𝑚+ ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 ² 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = 0 − ² 𝑣𝑣𝑣𝑣 4 𝑎𝑎𝑎𝑎 = 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = + ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 cht überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 4 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 4 𝑎𝑎𝑎𝑎 2 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 icht überschreiten. = 2 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚2 4 𝑎𝑎𝑎𝑎 2 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 4 𝑎𝑎𝑎𝑎 2 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 2 ∙ 𝑎𝑎𝑎𝑎 4 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 = 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚22∙ 2wenn die gewünschte Position innerhalb 𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= − 44𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +Anwendung, 𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝜋𝜋𝜋𝜋III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 nicht nicht überschreiten. überschreiten. 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Bewegungsgesetze =𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠 +𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 II 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎zur 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣2𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚 +𝑎𝑎𝑎𝑎42 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = +𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = ∙ 𝑡𝑡𝑡𝑡 2 −Diese 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 kommen bei translatorischen Bewegungen d 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III Bewegungsgesetz Parameter Weg Beschl 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑠𝑠𝑠𝑠 2= 𝑎𝑎𝑎𝑎+ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚²𝜋𝜋𝜋𝜋∙𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 2 −𝑎𝑎𝑎𝑎Geschwindigkeit 4 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 − 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 = 𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 4 𝑎𝑎𝑎𝑎 2 22 − 22 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 − 2𝑎𝑎𝑎𝑎 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 Geschwindigkeit Beschleunigung 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 = − 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4Beschleunigung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Sinus 22 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II zulässigen ² 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼zulässigen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋 =− 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III �1 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Dann �� wird 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 einer der und der Geschwindigkeit in𝑣𝑣𝑣𝑣 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 + 𝑣𝑣𝑣𝑣 führt. 𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 += �� ∙+ 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚zu 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+ Geschwindigkeit Beschleunigung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = + 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣− 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = == 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑎𝑎𝑎𝑎 III 𝜋𝜋𝜋𝜋𝐼𝐼𝐼𝐼22 ²= = 𝑠𝑠𝑠𝑠2𝑎𝑎𝑎𝑎 − + 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙Sinus −+ 𝑣𝑣𝑣𝑣 Sinus 𝑎𝑎𝑎𝑎Überschreitung 𝑣𝑣𝑣𝑣+ 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣Positionierzeit −𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣+ 𝑎𝑎𝑎𝑎 ∙2 𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡2III 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = −𝑎𝑎𝑎𝑎 𝜋𝜋𝜋𝜋Positionierzeit 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡 ²𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 ∙2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ∙∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼function �1 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼Konstantfahrtphase 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣zulässi ==𝑣𝑣𝑣𝑣� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 4𝑣𝑣𝑣𝑣= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sine 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡= 𝑠𝑠𝑠𝑠und − + ∙ �𝜋𝜋𝜋𝜋 + dass𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III𝑠𝑠𝑠𝑠∙∙𝑡𝑡𝑡𝑡Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑇𝑇𝑇𝑇 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣Anwendung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 − 𝑡𝑡𝑡𝑡 −𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 Bewegungsablaufs eine eingefügt die so verlängert, die𝑡𝑡𝑡𝑡Beschleunigung die− − ∙ 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼2𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎 2 𝑣𝑣𝑣𝑣 = ∙ 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 − 𝑎𝑎𝑎𝑎 = −𝑎𝑎𝑎𝑎 Sinus = 𝑠𝑠𝑠𝑠 − + ∙ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2 Geschwindigkeit 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 4 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� Sinus 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋 =die ∙𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+ 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 2 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ∙ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −−𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼Geschwindigkeit 4𝑠𝑠𝑠𝑠 =²𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 die Geschwindigkeit zulässige Geschwindigkeit nicht 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Geschwindigkeit ∙überschreiten. 𝑡𝑡𝑡𝑡42 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 =+ 𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit Beschleunigung Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼�𝑡𝑡𝑡𝑡 − �𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= + 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠=𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 III 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑠𝑠𝑠𝑠22𝑎𝑎𝑎𝑎 �𝜋𝜋𝜋𝜋 ��∙ �1 − 𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑣𝑣𝑣𝑣∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼−𝜋𝜋𝜋𝜋∙�𝑡𝑡𝑡𝑡 − + 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼² 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡 +𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣�𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠�𝜋𝜋𝜋𝜋 ++ 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 +== 𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 + 2 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣�� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 Sinus 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 Motion law Velocity Acceleration 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣= 2 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 222𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz Parameter Weg Geschwindigkeit Beschleun 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣2𝑎𝑎𝑎𝑎 4= 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = + 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣= =𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡 �𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣+=2𝑎𝑎𝑎𝑎 III∙𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣�� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎− 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 2Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 ∙ 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 2 22 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + 2 III 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼zulässige 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ==𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡das 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎==𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 keinen 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 2verlustoptimale mal Geschwindigkeit bestimmt, hat Bewegungsgesetz 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣=== 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎2𝑎𝑎𝑎𝑎==𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑣𝑣𝑣𝑣 Vorteil 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 2222 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =�𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃= 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚Bewegungsgesetz 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 das verlustoptimale 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 =𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣Vorteil 2 2 ch die maximal zulässige Geschwindigkeit bestimmt, hat keinen 2 eschwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz keinen Vorteil 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + malen Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale schwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz keinen Vorteil 22 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 lnus zulässige Geschwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz keinen Vorteil 2 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 zulässige Geschwindigkeit bestimmt, hat𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚das Bewegungsgesetz Vorteil 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎verlustoptimale Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22𝑎𝑎𝑎𝑎 = 20𝑎𝑎𝑎𝑎keinen 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 digkeitsoptimalen Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 = 0 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 ngsgesetz. Daher sollte indiesem diesem Fall auf−das das geschwindigkeitsoptimale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 lenBewegungsgesetz. Bewegungsgesetz. Daher sollteFall in diesem Fall auf dasgeschwindigkeitsoptimale geschwindigkeitsoptimale gsgesetz. sollte in auf geschwindigkeitsoptimale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Daher 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 diesem = 22 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 en Daher sollte in Fall auf das II 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣== 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎==22 00 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 en werden. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 + 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣==𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎==00 22 ++ 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼

Motion law Bewegungsgesetz

𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚zulässige Geschwindigkeit 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Wird der Bewegungsablauf durch die maximal bestimmt, hat das verlustoptimale Bewegungsges 2

22∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = + 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz. 2 Daher 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 dem geschwindigkeitsoptimalen in diesem Fall auf das geschwindigkeitsop 𝑣𝑣𝑣𝑣mehr = 𝑣𝑣𝑣𝑣gegenüber − 𝑎𝑎𝑎𝑎− ∙ 𝑡𝑡𝑡𝑡gegriffen 𝑎𝑎𝑎𝑎 = −𝑎𝑎𝑎𝑎 22 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑎𝑎𝑎𝑎sollte − ∙ 𝑡𝑡𝑡𝑡∙ 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝐼𝐼𝐼𝐼 2 + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −Time-optimised 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 werden. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz zurück 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙𝑣𝑣𝑣𝑣∙𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 ==𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚−−𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 ==−𝑎𝑎𝑎𝑎 ∙ ∙𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡 ∙2𝑡𝑡𝑡𝑡2− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 �2 − 𝑣𝑣𝑣𝑣 = 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎−𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =− ∙ �𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡= 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � �� 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 � =𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�1 ==𝑎𝑎𝑎𝑎−𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙2�1 𝐼𝐼𝐼𝐼 − 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼−− 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 222Verlustoptimal 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣− 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎bestimmt, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣Bewegungsablauf 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Wird der durch die maximal zulässige Geschwindigkeit hat das Bewegungsgesetz k 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙�𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣= = ∙ �1−−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡�𝐼𝐼𝐼𝐼�� 𝑣𝑣𝑣𝑣 verlustoptimale 𝐼𝐼𝐼𝐼�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �− 2𝑎𝑎𝑎𝑎 2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �� 𝑣𝑣𝑣𝑣��= = ∙�1 �1 −𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑎𝑎𝑎𝑎��= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣�� 𝑣𝑣𝑣𝑣2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼�� 𝑣𝑣𝑣𝑣 �� 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎� 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼�− ∙ − 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 ∙ 𝜋𝜋𝜋𝜋�� 𝑣𝑣𝑣𝑣∙𝑚𝑚𝑚𝑚�1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 geschwindigkeitsoptimalen 𝐼𝐼𝐼𝐼 mehr gegenüber dem Bewegungsgesetz. Daher sollte in diesem Fall 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼auf das geschwindigkeitsoptim 𝜋𝜋𝜋𝜋 2 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2= 𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ��𝑎𝑎𝑎𝑎 = 0 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑎𝑎𝑎𝑎 = 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 I 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22 Bewegungsgesetz 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2 𝑣𝑣𝑣𝑣 zurück gegriffen werden.

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = 0 𝑠𝑠𝑠𝑠 = +4𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎22 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋Geschwindigkeit 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚4𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =verlustoptimale − 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚keinen ige bestimmt, hat das Vorteil 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = 0 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚 Verlustoptimal 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = 0 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 = + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ² 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ige verlustoptimale Bewegungsgesetz = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = = 0Vorteil 0keinen Vorteil 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣das = 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 bestimmt, hat 𝑣𝑣𝑣𝑣 ∙𝑎𝑎𝑎𝑎∙𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4Geschwindigkeit Geschwindigkeit hat das verlustoptimale keinen =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ Bewegungsgesetz 4 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +bestimmt, ∙ bestimmt, hat das verlustoptimale Bewegungsgesetz keinen 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 ² 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼−𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣4𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Vorteil 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 das geschwindigkeitsoptimale wegungsgesetz. Daher sollte in diesem Fall auf 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣∙ 𝑚𝑚𝑚𝑚 ² 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 keinen ge eGeschwindigkeit Geschwindigkeit Geschwindigkeit bestimmt, hat hat das das verlustoptimale Bewegungsgesetz 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2bestimmt, 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼verlustoptimale = = 𝑠𝑠𝑠𝑠 = �𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠keinen � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼Vorteil ��Vorteil 𝑣𝑣𝑣𝑣 =𝑣𝑣𝑣𝑣2𝑎𝑎𝑎𝑎 �1 ∙− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I Bewegungsgesetz 𝐼𝐼𝐼𝐼= wegungsgesetz. Daher sollte in diesem auf geschwindigkeitsoptimale 𝐼𝐼𝐼𝐼 − + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Fall ∙2 das + ∙ =2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −2 ∙ 2das 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 ungsgesetz. Daher sollte in diesem Fall auf geschwindigkeitsoptimale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale 2 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑣𝑣𝑣𝑣 2 ² 𝑣𝑣𝑣𝑣 4 𝑎𝑎𝑎𝑎 2 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ² 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 egungsgesetz. gungsgesetz. Dahersollte sollteinindiesem diesem Fall Fall auf auf das das geschwindigkeitsoptimale geschwindigkeitsoptimale 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ Daher 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇 = + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = ∙ 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 == 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 − ∙2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 2 ∙ Sinus 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = II 𝑠𝑠𝑠𝑠 = 𝑡𝑡𝑡𝑡 �� +𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 ∙= ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 4𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 function − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= �𝜋𝜋𝜋𝜋 +2 ∙∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣�� 1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣Sine 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝜋𝜋𝜋𝜋+ + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ² 𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋𝑠𝑠𝑠𝑠 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙∙2𝑎𝑎𝑎𝑎 = �𝜋𝜋𝜋𝜋 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = � ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑎𝑎𝑎𝑎 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎4 = =𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎�𝜋𝜋𝜋𝜋 �𝜋𝜋𝜋𝜋 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼2𝑣𝑣𝑣𝑣� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 𝑇𝑇𝑇𝑇 = + 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 �1+2− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + �� 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼��and velocity 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼−− 2𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + laws𝑡𝑡𝑡𝑡of 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼translation ��𝑣𝑣𝑣𝑣−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣�𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠∙�𝜋𝜋𝜋𝜋 Table 15𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋 with limitation of acceleration 𝑣𝑣𝑣𝑣22 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 + 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝜋𝜋𝜋𝜋 �𝜋𝜋𝜋𝜋 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Motion �� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 2𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 = �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 � 𝐼𝐼𝐼𝐼�� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�� 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋∙ + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 −= 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� = ∙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �1�− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎 ∙= 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡��𝐼𝐼𝐼𝐼 � 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣== 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙2∙�1 ��𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑠𝑠𝑠𝑠 �1−−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝑎𝑎𝑎𝑎 �� 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎==𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 �𝑣𝑣𝑣𝑣 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣==22 ∙ 2 ∙�1 �1−−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 �𝑣𝑣𝑣𝑣 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎==𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 �𝑣𝑣𝑣𝑣 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣�𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�� 𝐼𝐼𝐼𝐼 �� 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡�� 𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 2 2 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚28 22𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = 0 22

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Tabelle 1515 Bewegungsgesetze derder Translation mitmit Begrenzung derder Beschleunigung und derder Geschwindigkeit Tabelle Bewegungsgesetze Translation Begrenzung Beschleunigung und Geschwindigkeit Tabelle 15 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung und der Geschwindigkeit abelle 15 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung und der Geschwindigkeit Rotation, Vorgabewerte: ϕPos , TPos Description of the load cycle Tabelle 15Vorgabewerte: Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung und der Geschwindigkeit ϕPos ,T Rotation, ϕPos ,Pos TDiese Rotation, Vorgabewerte: Pos Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die g Tabelle 15 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung und der Geschwindigkeit Anwendung Tabelle 15 Bewegungsgesetze der Translation ohne mit Begrenzung derzuBeschleunigung und der Geschwindigkeit ϕPos ,Diese TPositionierzeit Rotation, Vorgabewerte: Pos ϕPos ,T Rotation, Vorgabewerte: Überschreitung zulässigenBewegungen Winkelgeschwindigkeit und der zulässigen Pos Bewegungsgesetze kommen rotatorischen zur Anwendung, wenn diedie gew elle 15 Bewegungsgesetze der Translation mit Begrenzung derbei Beschleunigung und der Geschwindigkeit Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn g Tabelle 15 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung und der Geschwindigkeit Anwendung Anwendung ,Positionierzeit TPositionierzeit Rotation, Vorgabewerte: These motion lawsϕare movements if the position can be realizedund within Pos used Posin rotating ohne Überschreitung zudesired zulässigen Winkelgeschwindigkeit derder zulässigen W ohne Überschreitung zu zulässigen Winkelgeschwindigkeit und zulässigen Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die ge Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die gewü Anwendung ,T Rotation, Vorgabewerte: Pos Pos Anwendung Bewegungsgesetz Parameter Weg the Vorgabewerte: required positioning time without exceeding the permissible angular speed and permisϕPosPositionierzeit ,ϕT Rotation, Pos Positionierzeit ohne Überschreitung zurotatorischen zulässigen Winkelgeschwindigkeit und der zulässigen ohne Überschreitung zu Winkelgeschwindigkeit und derGeschwindigk zulässigen Diese Bewegungsgesetze kommen beizulässigen Bewegungen zurthe Anwendung, wenn dieWin ge ϕPos ,ϕTPos tation, Vorgabewerte: Anwendung Pos,Parameter TPositionierzeit Rotation, Vorgabewerte: Bewegungsgesetz Weg Geschwindigkei Pos Bewegungsgesetz Parameter Weg Geschwindigk ohne Überschreitung zu zulässigen Winkelgeschwindigkeit und der zulässigen W Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn sible angular acceleration. Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die ge Anwendung Translation, Vorgabewerte: sPos , TPos Anwendung ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 kommen ϕBewegungen ϕ Positionierzeit ohne Überschreitung zu zulässigen Winkelgeschwindigkeit und der zuläss Bewegungsgesetz Parameter Weg Geschwindigke Diese Bewegungsgesetze bei rotatorischen Bewegungen zur Anwendung, wenn die gewünsc Bewegungsgesetz Parameter Weg Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese Bewegungsgesetze kommen bei rotatorischen zur Anwendung, wenn die ge 2 Positionierzeit ohne Überschreitung zu zulässigen und der zulässigen wendung ϕ = ϕ2 Winkelgeschwindigkeit ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 = ϕ4 W Translation, Vorgabewerte: sPos ,αT𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Pos = 4 I Anwendung 2 2zur ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Positionierzeit ohne Überschreitung zulässigen Winkelgeschwindigkeit und zulässigen Winke Diese kommenWeg beizu translatorischen Bewegungen wennder die gewünschte Position in𝑃𝑃𝑃𝑃W Bewegungsgesetz Parameter Geschwindigke Positionierzeit ohne Überschreitung zu zulässigen Winkelgeschwindigkeit und der zulässigen 𝑇𝑇𝑇𝑇ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇ϕ ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Translation, Vorgabewerte: , TPosBewegungsgesetze 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Pos 2 Anwendung, Anwendung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 zulässigen �sPos αPositionierzeit = 4 ϕ = 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Rotation, settings:: , T ohne Überschreitung zu zulässigen Geschwindigkeit und der Beschleunigung realisiert Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position 2 2 α = 4 ϕ = 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 4 =𝑇𝑇𝑇𝑇werd 4 in I Pos 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Bewegungsgesetz Parameter Weg Geschwind 2 2 Anwendung 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑇𝑇𝑇𝑇ohne ϕ𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕwerde Diese Bewegungsgesetze kommenWeg beizu translatorischen Bewegungen zur der Anwendung, wenn die gewünschte Position inn 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter Geschwindigke Positionierzeit zulässigen Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Überschreitung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Anwendung 2 2 zulässigen Beschleunigung realisiert α𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 α𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 4= 4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =ϕ2= 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡2und 𝜔𝜔𝜔𝜔 =𝜔𝜔𝜔𝜔4=Beschl wegungsgesetz Parameter Geschwindigkeit IWeg SegPositionierzeit ohne zu zulässigenϕGeschwindigkeit realisiert werde I Weg 𝐼𝐼𝐼𝐼 zulässigen Beschleunigung 𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡der 2 Überschreitung Bewegungsgesetz Parameter Weg ϕ2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter GeschwindigkeitGeschwindigke 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 Motion law Parameter Distance 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕ α ϕ =ϕ2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ϕGeschwindigkeit 𝜔𝜔𝜔𝜔 = 4 𝑃𝑃𝑃𝑃 IWeg ment 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 4 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter Beschle 2ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇− ϕϕ = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +ϕ2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − ϕ2 Geschwindigkeit 𝜔𝜔𝜔𝜔 =ϕ2Beschle 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =ϕ𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Weg III 2 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz Parameter 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ= ϕ α 4 ϕ = 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔ϕ= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 I 2𝑠𝑠𝑠𝑠 ϕϕ=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 I 𝑇𝑇𝑇𝑇ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 2 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 α𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ϕ== 4𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 4− ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ4−𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 + 2 ϕ = ∙ ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 2 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃III 2− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 + 2 ϕ = ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 2 = 𝑠𝑠𝑠𝑠 2 = 2=𝑠𝑠𝑠𝑠 2ϕ ∙= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼= 4 𝑠𝑠𝑠𝑠 2 ∙ 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 = 4𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 =24𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇2 𝑇𝑇𝑇𝑇 4𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 III I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 I 2 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 α𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =4ϕ = 4 𝑇𝑇𝑇𝑇 Geschwindigkeitsoptimal α𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 2 I 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 2 2 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ϕ 𝑠𝑠𝑠𝑠 ϕ ϕ𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑇𝑇𝑇𝑇ϕ 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 4 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡ϕ2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +22𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2∙ 𝑡𝑡𝑡𝑡ϕ𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡− ∙ 2𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼=2 4 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔2= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃4 𝜔𝜔𝜔𝜔 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ =ϕ =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 4𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = =2=4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III III 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2− 2 ϕ2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 ϕ =2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal I 2 ϕ Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +2 ϕ = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 −4 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 III 2ϕ ϕ 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇 Geschwindigkeitsoptimal Geschwindigkeitsoptimal 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕ ϕ ϕ𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑠𝑠𝑠𝑠 = + 2 − ∙ 𝑡𝑡𝑡𝑡 2 − 4 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 2 III 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ϕ = + 2 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 2− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 𝑠𝑠𝑠𝑠ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕ + 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ 𝑡𝑡𝑡𝑡 ϕ− 𝑠𝑠𝑠𝑠2𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜔𝜔𝜔𝜔ϕ = 2𝜔𝜔𝜔𝜔 = 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = III 𝑠𝑠𝑠𝑠6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 ϕ 2 ϕ = 2 ∙ 𝑡𝑡𝑡𝑡 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 2 Time-optimised 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃+ 2ϕ𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 4 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ϕ ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2−ϕ2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇= 2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 Geschwindigkeitsoptimal 𝑣𝑣𝑣𝑣 = 2 𝜑𝜑𝜑𝜑 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 Geschwindigkeitsoptimal𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑠𝑠𝑠𝑠 = = 2ϕ +=2+𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡+ 𝑡𝑡𝑡𝑡2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −2∙ 2 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃22 𝜔𝜔𝜔𝜔 4 −𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝛼𝛼𝛼𝛼 2 =6 III −∙ 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2− 2 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃−𝜔𝜔𝜔𝜔 4 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇 = 2−𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 2𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚== 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 III III =2𝑇𝑇𝑇𝑇 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 = 3 ∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 6 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 3 2 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6=66 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 schwindigkeitsoptimal 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal 2 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 = 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 3 =𝑇𝑇𝑇𝑇3 2 ∙ 2𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡−2 2−𝑇𝑇𝑇𝑇2 3 ∙ 3𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 3 𝜔𝜔𝜔𝜔 6 =𝑇𝑇𝑇𝑇6 2 ∙ 2𝑡𝑡𝑡𝑡 ∙−𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 3 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑 2 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎 = 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 er 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 erBeschleunigung Beschleunigungund undder derGeschwindigkeit Geschwindigkeit 2 𝜑𝜑𝜑𝜑 3 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2∙ 3𝑡𝑡𝑡𝑡∙2𝑡𝑡𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 =𝜑𝜑𝜑𝜑3 = ∙ 𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 6 3𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎6− 𝑠𝑠𝑠𝑠𝜑𝜑𝜑𝜑 𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔 2 ∙ 𝑡𝑡𝑡𝑡3 − 2∙ 𝑠𝑠𝑠𝑠 3 ∙6 3= 𝑡𝑡𝑡𝑡2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 23 2 𝜑𝜑𝜑𝜑3 ∙ 𝑡𝑡𝑡𝑡33∙ 𝑡𝑡𝑡𝑡 3 𝑣𝑣𝑣𝑣 = 6 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔6= 6= ∙ 𝑡𝑡𝑡𝑡2 − 2 2𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑= 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃6 3 𝜑𝜑𝜑𝜑2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 2 𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 3 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡 2 𝑇𝑇𝑇𝑇 − 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 = Verlustoptimal 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 ∙2𝑡𝑡𝑡𝑡3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑇𝑇𝑇𝑇 3 𝑣𝑣𝑣𝑣 = 6 𝑠𝑠𝑠𝑠 3 𝑠𝑠𝑠𝑠 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔= 6 𝜔𝜔𝜔𝜔 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝜔𝜔𝜔𝜔 ∙ 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡 =− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 3 𝜑𝜑𝜑𝜑 ∙ 𝑡𝑡𝑡𝑡3 − 2 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙3𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡 − 2 ∙ 𝑡𝑡𝑡𝑡6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3= 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 3 2 3 𝑠𝑠𝑠𝑠 2 𝑇𝑇𝑇𝑇 3𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3= 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 er Beschleunigung undder derGeschwindigkeit Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 r Verlustoptimal Beschleunigung und Verlustoptimal 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =322 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑= 3 ∙ 𝑡𝑡𝑡𝑡 2 − ∙ 𝑡𝑡𝑡𝑡 2𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 2 ∙ 𝑡𝑡𝑡𝑡 3 3 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔𝜑𝜑𝜑𝜑= 6 ∙ 𝑡𝑡𝑡𝑡 − 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 == Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑= 𝜔𝜔𝜔𝜔𝜑𝜑𝜑𝜑= 𝜑𝜑𝜑𝜑2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃6 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 Loss-optimised 2 Verlustoptimal 2 3 𝜔𝜔𝜔𝜔 = Verlustoptimal 𝜑𝜑𝜑𝜑 = 3𝜑𝜑𝜑𝜑 = 23 𝑇𝑇𝑇𝑇 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 − ∙2𝑡𝑡𝑡𝑡 − 32 𝑇𝑇𝑇𝑇 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 6𝜔𝜔𝜔𝜔 = 26 𝑇𝑇𝑇𝑇 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 − 6∙ 𝑡𝑡𝑡𝑡 − Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 3 𝜑𝜑𝜑𝜑 2 3 ∙ 𝑡𝑡𝑡𝑡 2 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 chleunigung und der Geschwindigkeit 2 𝑇𝑇𝑇𝑇 Beschleunigung und der Geschwindigkeit schleunigung und der Geschwindigkeit chleunigung und der Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 rotatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten 3 𝑠𝑠𝑠𝑠𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚=𝜔𝜔𝜔𝜔 = 2𝜋𝜋𝜋𝜋 i rotatorischen Bewegungen zur Anwendung, wenn die der geforderten 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal = 2gewünschte Position innerhalb 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝜑𝜑𝜑𝜑=zulässigen 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋3𝜑𝜑𝜑𝜑 2 𝑇𝑇𝑇𝑇Winkelbeschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 u zulässigen Winkelgeschwindigkeit und der realisiert werden kann. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 zu zulässigen Winkelgeschwindigkeit und der zulässigen Winkelbeschleunigung realisiert werden kann. 𝛼𝛼𝛼𝛼 = 2𝜋𝜋𝜋𝜋 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = Verlustoptimal 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 == 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 =2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 22 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇innerhalb 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑2𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 rotatorischen Bewegungenzur zurAnwendung, Anwendung, die gewünschte Position der geforderten = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 =Position 𝑡𝑡𝑡𝑡��geforderten lustoptimal 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑∙ �𝑡𝑡𝑡𝑡=−innerhalb 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋= rotatorischen Bewegungen wenn die gewünschte 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � −der 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 ∙ �1𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 Verlustoptimal 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =wenn 2𝜋𝜋𝜋𝜋𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 zulässigenWinkelgeschwindigkeit Winkelgeschwindigkeitund und zulässigen Winkelbeschleunigung realisiert werden kann. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 Geschwindigkeit Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 = 2 𝑣𝑣𝑣𝑣der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 uuzulässigen der zulässigen Winkelbeschleunigung realisiert werden kann. 𝑎𝑎𝑎𝑎 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝜑𝜑𝜑𝜑 = gewünschte 2 𝑠𝑠𝑠𝑠gewünschte 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇innerhalb 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 rischen zur Anwendung, wenn die innerhalb der Translation, Vorgabewerte: sPos , 𝑣𝑣𝑣𝑣T𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 otatorischen Bewegungen zur Anwendung, wenn die gewünschte Position der geforderten orischen Bewegungen zur Anwendung, wenn die Position innerhalb dergeforderten geforderten Pos 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 rischenBewegungen Bewegungen zur Anwendung, wenn die Position innerhalb geforderten 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Position 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃der𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =2gewünschte 2𝑇𝑇𝑇𝑇= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 Sinus 𝑠𝑠𝑠𝑠 � kann. =realisiert ∙−�𝑡𝑡𝑡𝑡werden − 𝑡𝑡𝑡𝑡�� ∙− �1𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 − 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �kann. 𝜑𝜑𝜑𝜑bei=𝜑𝜑𝜑𝜑 ∙ �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 gewünschte =𝜔𝜔𝜔𝜔 = 𝜑𝜑𝜑𝜑 ∙ �1 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 2𝜋𝜋𝜋𝜋 sigen und der zulässigen Winkelbeschleunigung kann. 𝜑𝜑𝜑𝜑 zulässigen Winkelgeschwindigkeit und der zulässigen Winkelbeschleunigung realisiert werden ssigen Winkelgeschwindigkeit und der zulässigen Winkelbeschleunigung realisiert werden kann. 𝑇𝑇𝑇𝑇 sigenWinkelgeschwindigkeit Winkelgeschwindigkeit und der zulässigen Winkelbeschleunigung realisiert werden 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 Sinus 2 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇 Diese Bewegungsgesetze kommen translatorischen Bewegungen zur Anwendung, wenn die Translation, Vorgabewerte: s , T 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Pos Pos Sine function 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Position in 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ϕ ϕ 𝜔𝜔𝜔𝜔 = 2 13 Bewegungsgesetze der Translation ohne Begrenzungen SinusTabelle 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 = 2𝜋𝜋𝜋𝜋 Anwendung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = 2𝜋𝜋𝜋𝜋 2 2 Vorgabewerte: sPosPositionierzeit 𝜔𝜔𝜔𝜔 2 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝑐𝑐𝑐𝑐 , T 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Pos =𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ohne Überschreitung zu zulässigen Geschwindigkeit und der zulässigen Beschleunigung realisiert werd 2 ϕϕ==2Tabelle ∙ 𝑡𝑡𝑡𝑡 = 4 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 4 𝑇𝑇𝑇𝑇 2Translation, ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 4 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 4 13 Bewegungsgesetze der Translation ohne Begrenzungen 𝑇𝑇𝑇𝑇Bewegungsgesetze 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2𝜋𝜋𝜋𝜋 wenn die gewünschte 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 in 2 2 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2 2kommen 2 2Anwendung, 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese bei translatorischen Bewegungen zur Position 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus der Translation ohne Begrenzungen 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Anwendung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃− 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 13 Bewegungsgesetze 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus Tabelle 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 �zulässigen 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∙Geschwindigkeit ∙𝑇𝑇𝑇𝑇�𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∙ �1werde ∙ 𝑐𝑐𝑐𝑐 𝜔𝜔𝜔𝜔 = 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ohne 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2Bewegungsgesetze Diese kommen bei𝜑𝜑𝜑𝜑 translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position inn Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit Beschleunigung Geschwindigkeit Beschleunigung Geschwindigkeit Beschleunigung Positionierzeit Überschreitung zu zulässigen und der Beschleunigung realisiert 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝜑𝜑𝜑𝜑 = �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = − 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 2 Motion law Velocity Acceleration 𝜑𝜑𝜑𝜑 Anwendung 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇Geschwindigkeit 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Beschl 2𝜋𝜋𝜋𝜋 zulässigen ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕϕÜberschreitung 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Weg 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Positionierzeit zulässigen Beschleunigung werde 𝑇𝑇𝑇𝑇�1 2 𝑠𝑠𝑠𝑠 = � 4 − Geschwindigkeit 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 =∙realisiert − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔Parameter 2𝜔𝜔𝜔𝜔==4ohne Sinus 𝑠𝑠𝑠𝑠und �𝑇𝑇𝑇𝑇der =∙ 𝑇𝑇𝑇𝑇�𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼2𝜋𝜋𝜋𝜋 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝜑𝜑𝜑𝜑 = zu ϕϕ==22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 2∙ 𝑡𝑡𝑡𝑡∙ 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 =4 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 −𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃4 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2𝑇𝑇𝑇𝑇2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝛼𝛼𝛼𝛼 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 Bewegungsgesetz Parameter Weg Geschwindigkeit Beschle 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 23 𝜔𝜔𝜔𝜔 2 ϕ ϕ ϕ 𝜑𝜑𝜑𝜑 Bewegungsgesetz Parameter Weg Geschwindigkeit Beschle ϕ ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ ϕ ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠ϕ 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜔𝜔𝜔𝜔 2= = ∙ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4𝜔𝜔𝜔𝜔 = Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ+2+= 4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 42−4 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡2222𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 ∙𝑡𝑡𝑡𝑡− =4𝛼𝛼𝛼𝛼 ==22 𝜔𝜔𝜔𝜔 == 44224= 𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 44 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔= = 𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 22 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡− 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 = 2𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = −4 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎2 = 𝑠𝑠𝑠𝑠 = 2𝛼𝛼𝛼𝛼 𝑣𝑣𝑣𝑣 = 4 𝐼𝐼𝐼𝐼 −22 𝐼𝐼𝐼𝐼24 𝐼𝐼𝐼𝐼 4 2222∙∙− I∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 2∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 232 2 2 22∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼 ∙𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 223 2 2∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 202 ∙ 𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22Sinus 𝑇𝑇𝑇𝑇2𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃20 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 usSinus 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 4 ϕ 𝑠𝑠𝑠𝑠 = 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 202 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 Iϕ 23 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 4ϕ𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 23 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 − 4I 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼−−22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 2 𝜔𝜔𝜔𝜔==22𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼==−4 −4 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 + ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 − 4 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 2 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 23 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 2 −4 𝑣𝑣𝑣𝑣ϕ =ϕ2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ϕ𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕϕϕ ϕ ϕ ϕ ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ ϕ ϕ ϕ ϕ ϕ ϕ 23 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Time-optimised 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 222 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠−4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 23 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼∙= −4 ∙− 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 = 2=− ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼− − 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 4 − 4222∙∙III𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −4 22 + 2 ∙∙ 𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22 − 2222∙∙ 𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= ==2𝜔𝜔𝜔𝜔 22𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −2−4𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠4𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = −4 Geschwindigkeitsoptimal 𝑡𝑡𝑡𝑡= − 2 𝑣𝑣𝑣𝑣 = 2𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 4 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22∙2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2 2 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 23 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇23𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 2 + 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣 = 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃− 4 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇 =𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeitsoptimal Geschwindigkeitsoptimal

2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔==66 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼==66 2 2−−12 −66 3 3∙ ∙𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 12 3 3∙ ∙𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ ∙𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡− 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇 6𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃222 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 2 − 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 2 3 2 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑3𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡−−66 ∙ 2= 𝑡𝑡𝑡𝑡 23 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡22𝛼𝛼𝛼𝛼− 𝛼𝛼𝛼𝛼=2=𝑠𝑠𝑠𝑠6𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 6 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 12 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3∙𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡∙ =𝑡𝑡𝑡𝑡 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡33 2−−12 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 ==6=6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 ∙ 𝑡𝑡𝑡𝑡2 2∙ 𝑡𝑡𝑡𝑡 3∙ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 3 2 3 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡 − 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 ∙ 𝑡𝑡𝑡𝑡 − 6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 ∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡− 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 622= − 12333∙∙𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡3 ∙ 𝑡𝑡𝑡𝑡 −6∙6𝑡𝑡𝑡𝑡6 − 6333∙∙𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡232 ∙ 𝑡𝑡𝑡𝑡 2 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= ==6𝜔𝜔𝜔𝜔 66= −𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ==6𝛼𝛼𝛼𝛼 66= 6222− −−12 12 212 2∙ ∙2 𝑇𝑇𝑇𝑇2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 2 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑣𝑣𝑣𝑣 𝜑𝜑𝜑𝜑= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡� 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼= 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��2 2∙ ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�� 𝑡𝑡𝑡𝑡�� ∙ ∙�𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡−− 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 == ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙�1�1−−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠�� 𝑠𝑠𝑠𝑠 =𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠2𝜋𝜋𝜋𝜋 �2𝜋𝜋𝜋𝜋 =𝑡𝑡𝑡𝑡�𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 13 der Translation ohne Begrenzungen Sinus ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�� 𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡� 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � Bewegungsgesetze 𝑡𝑡𝑡𝑡�� = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 ∙ �1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼==2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�function ∙ �𝑡𝑡𝑡𝑡 − Tabelle 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= −−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 ��𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡�� Sine Sinus 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇Bewegungsgesetze 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ohne 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Tabelle 13 der Translation Begrenzungen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 13 Bewegungsgesetze𝜑𝜑𝜑𝜑 der Translation ohne Begrenzungen 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Tabelle 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 33 2 2∙ ∙𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2 2−−22 3 3∙ ∙𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡3 3 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2∙ 𝑡𝑡𝑡𝑡∙ 2𝑡𝑡𝑡𝑡 2−−22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3∙ 𝑡𝑡𝑡𝑡∙ 3𝑡𝑡𝑡𝑡 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Verlustoptimal 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 222 2 ∙∙ 𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡22 −Loss-optimised 2333∙∙𝑡𝑡𝑡𝑡∙𝑡𝑡𝑡𝑡3𝑡𝑡𝑡𝑡333 ∙ 𝑡𝑡𝑡𝑡 3 −∙−2 𝑇𝑇𝑇𝑇2 2 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 Verlustoptimal

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠��𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡�� − 𝜔𝜔𝜔𝜔 ∙ �𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �1 −��𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� ∙�1 �1− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � �𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 −− 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= ==𝜔𝜔𝜔𝜔 = ∙∙�1 −∙−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 limitations 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 Table 16𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Motion laws of rotation without 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 23 23

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡� 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼= 𝛼𝛼𝛼𝛼 = 2𝜋𝜋𝜋𝜋222∙∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡� 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡� ==2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝑠𝑠𝑠𝑠∙�𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 20

𝑎𝑎𝑎𝑎 =

𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 =

𝑎𝑎𝑎𝑎 =

𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 =

Description of ϕ the cycle otation, Vorgabewerte: , αmax Posload

SegSegSegment ment ment

Seg-SegSegment ment ment

SegmentSegment

SegSeg-SegSegment ment SegSeg- ment SegSegment mentment Segment ment ment

Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die ge nwendung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt.ofDann wird on, Vorgabewerte: ϕPos , laws αmaxϕare Rotation, , αmaxin rotat TheseVorgabewerte: motion movements if the desired position leads to an exceeding Posused Rotation, Vorgabewerte: ϕPos , αmax Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht überschreitet. maximale the permissible angular acceleration within the required positioning time. Then the positioning Rotation, Vorgabewerte: ,αDiese α Pos maxBewegungsgesetze Rotation, Vorgabewerte: ϕϕ ,α Pos max Rotation, Rotation, Vorgabewerte: Vorgabewerte: ϕ ,,Bewegungsgesetze α ,αα Vorgabewerte: ϕPos , αmax Pos Pos max max bei rotatorischen Bewegungen zur Anwendung, wenn diegewün gewün Rotation, Vorgabewerte: ϕ ,max Pos max Diese kommenkommen bei rotatorischen Bewegungen zur Anwendung, wenn die Rotation, Vorgabewerte: ϕϕ Pos Anwendung Diese Bewegungsgesetze bei rotatorischen Bewegungen zurthe Anwendung, wenn diewird gewünsch Positionierzeit zuangular einerkommen Überschreitung der does zulässigen führt. Dann wirddie die time is extended soParameter that the maximum acceleration not Winkelbeschleunigung exceed permissible ndung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt. Dann ewegungsgesetz Weg Geschwind Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zurzur Anwendung, wenn die gewün Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn Anwendung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt. Dann wird diedie Po maximale Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht überschreitet. Diese Diese Bewegungsgesetze Bewegungsgesetze kommen kommen bei bei rotatorischen rotatorischen Bewegungen Bewegungen zur Anwendung, Anwendung, wenn wenn di Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die gewünsch Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn d Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die gewüns maximale Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht überschreitet. angular acceleration. Anwendung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt.führt. Dann wird die maximale Winkelbeschleunigung diezulässigen zulässige Winkelbeschleunigung nicht überschreitet. Anwendung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt. Dann Anwendung Anwendung ng Positionierzeit Positionierzeit zuzu einer einer Überschreitung Überschreitung der zulässigen zulässigen Winkelbeschleunigung Winkelbeschleunigung führt. Dann Dann wPw Positionierzeit zu einer Überschreitung der Winkelbeschleunigung führt. Dann wird die Pos Anwendung Anwendung Positionierzeit zu einer Überschreitung derder zulässigen Winkelbeschleunigung führt. Dann wird die Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung führt. Dann maximale Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht überschreitet. Translation, Vorgabewerte: smaximale ,Parameter amaximale Pos max Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht überschreite Bewegungsgesetz Weg Geschwindigk maximale maximale Winkelbeschleunigung die zulässige zulässige Winkelbeschleunigung Winkelbeschleunigung nicht nicht überschreitet. überschreitet. maximale Winkelbeschleunigung die zulässige Winkelbeschleunigung nicht nicht überschreitet. Winkelbeschleunigung diedie zulässige Winkelbeschleunigung überschreitet. ϕWinkelbeschleunigung maximale Winkelbeschleunigung überschreitet. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 gungsgesetz Parameter WegWegdie zulässige Winkelbeschleunigung nicht Geschwindigk Bewegungsgesetz Parameter Geschwindigkeit Translation, ss𝑇𝑇𝑇𝑇 ,= Pos 𝑇𝑇𝑇𝑇αBewegungsgesetze Translation, Vorgabewerte: Vorgabewerte: , aa,max 𝐼𝐼𝐼𝐼 Diese 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = � � Pos max kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb Rotation, settings: 𝛼𝛼𝛼𝛼 Pos max zu einer Bewegungsgesetz Parameter Weg Beschleunigung𝛼𝛼𝛼𝛼führt. Geschwindigk 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Überschreitung der zulässigen Anwendung Positionierzeit die Positionierzeit so verlängert, dass 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Dann2 wird Bewegungsgesetz Parameter Weg Gesch Bewegungsgesetz Parameter Weg Geschwindigke Diese kommen Bewegungen Position innerhalb ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Bewegungsgesetz Parameter Parameter Weg Weg Geschwi Geschw gsgesetz Parameter Weg Geschwindigkeit Bewegungsgesetz Parameter Weg Geschw ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 wenn Ibei ϕ =zur DieseBewegungsgesetze Bewegungsgesetze kommen beitranslatorischen translatorischen Bewegungen zurAnwendung, Anwendung, wenndie diegewünschte gewünschte Position innerhal Beschleunigung die zulässige Beschleunigung nicht überschreitet. Seg𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 =ϕ𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼ϕ =einer Anwendung � Positionierzeit zu Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positionierzeit so 2 Motion law Parameter Angle ϕ Anwendung Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positionierzeit soverlängert, verlängert,dass das 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ment 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Beschleunigung die 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼=== 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Beschleunigung Beschleunigung diezulässige zulässige Beschleunigungnicht nicht überschreitet. überschreitet. �𝑇𝑇𝑇𝑇�4 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = � 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsgesetz Parameter Weg I Geschwindigkeit Besch ∙ 𝑡𝑡𝑡𝑡 ϕ = 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =ϕϕ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 2 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝛼𝛼𝛼𝛼ϕ ϕ𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝜔𝜔𝜔𝜔Besch == 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =𝑇𝑇𝑇𝑇= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � Bewegungsgesetz 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝐼𝐼𝐼𝐼 == 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 Geschwindigkeit I Weg ϕ =ϕ = 2𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =Parameter Parameter Weg Geschwindigkeit Besch 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇= 2 � � � = 𝑇𝑇𝑇𝑇= = = 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇= 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I ϕ2𝛼𝛼𝛼𝛼 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 = 𝑠𝑠𝑠𝑠ϕ 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇ϕ =�4 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I ϕ = 2 ϕ2 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 � ϕ𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2∙ 2𝑡𝑡𝑡𝑡2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇�4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼∙𝑡𝑡𝑡𝑡− I I I ϕII = ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =+ �2ϕ ∙ϕ𝑡𝑡𝑡𝑡ϕ∙𝐼𝐼𝐼𝐼= 𝜔𝜔𝜔𝜔 = =� 𝛼𝛼𝛼𝛼ϕ 2𝛼𝛼𝛼𝛼=== 𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 2 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑠𝑠𝑠𝑠 �4 ϕ 𝜔𝜔𝜔𝜔 = ϕ ∙ 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eschwindigkeitsoptimal ϕI ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 2 22 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 =� �𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝛼𝛼𝛼𝛼ϕ 𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2ϕ𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇ϕ 2 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃= 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+ 2 2 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 = = 𝑇𝑇𝑇𝑇 = ∙ 𝑡𝑡𝑡𝑡 I 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇 = 2 �4 �4 𝜔𝜔𝜔𝜔 ϕ ∙ 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 = ϕ ∙ �4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �4 III 𝑇𝑇𝑇𝑇 = ϕ = ϕ ∙ 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 �4 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 � � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼 Geschwindigkeitsoptimal I = 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ϕ𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �ϕ 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 = �ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚 III ϕ +2 �ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ = ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃== Geschwindigkeitsoptimal �4 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �4𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 �ϕ + � 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �ϕ𝜔𝜔𝜔𝜔𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔==�� ϕϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ ∙𝛼𝛼𝛼𝛼 ϕ ϕ=+ ∙ 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙−𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − III ϕIII𝑠𝑠𝑠𝑠=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ ϕ𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �ϕ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ = Geschwindigkeitsoptimal 𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 hwindigkeitsoptimal 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 = + ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 ∙ 𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 ϕ 2 2 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔∙ = 𝜔𝜔𝜔𝜔 = �6 III ϕ𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 +ϕ�𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙ 𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 �ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼ϕ𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2ϕ = 2 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 2 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜔𝜔𝜔𝜔∙ 𝛼𝛼𝛼𝛼 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ϕ ∙ 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 2 ∙ 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ = + ϕ ∙ 𝑡𝑡𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 2 2 2 � 2 2 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal Time-optimised 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 III 𝑠𝑠𝑠𝑠 = + ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝜔𝜔𝜔𝜔 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ϕ �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 =𝑃𝑃𝑃𝑃� ϕ ∙ 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔= =�=� Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙∙ 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = = ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃� ϕ∙∙�6 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔∙𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �ϕ ∙ 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III ++ϕ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃ϕ== +−� III ϕ = + + ϕ�𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 ∙ ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣∙ ∙= 𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −− ∙ −𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡−∙∙𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼 ∙ϕ2ϕ𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − III ϕ = 𝑠𝑠𝑠𝑠 III =III ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 � �𝑠𝑠𝑠𝑠 � � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇=𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal Geschwindigkeitsoptimal ndigkeitsoptimal Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2𝛼𝛼𝛼𝛼 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 222 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 1 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 ϕ = �6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ϕ = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑡𝑡𝑡𝑡 22− 1 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡33 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �6 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 1− 3 ϕ= 2 ∙ 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = �6 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 �6 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝛼𝛼𝛼𝛼1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝛼𝛼𝛼𝛼 𝑎𝑎𝑎𝑎 2 𝑇𝑇𝑇𝑇 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ϕ �6 2 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Position ∙ 𝑡𝑡𝑡𝑡∙geforderten ϕ∙ 𝑡𝑡𝑡𝑡=− ∙ 𝛼𝛼𝛼𝛼der − ∙ 1 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑎𝑎𝑎𝑎∙ =𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = innerhalb 𝑡𝑡𝑡𝑡geforderten rischen Bewegungen gewünschte orischen Bewegungenzur zurAnwendung, Anwendung,wenn wenn gewünschte der ϕ ϕϕ ϕ �die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Position 2innerhalb 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3𝛼𝛼𝛼𝛼 23𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙1 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1312𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔 =𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ�6 ∙�6 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 3𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2ϕ = 3𝑡𝑡𝑡𝑡 − �6 𝑇𝑇𝑇𝑇 == 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 führt. = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 22 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �6Dann 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 verlängert, 𝛼𝛼𝛼𝛼 ��6 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = ∙𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡verlängert, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = ∙∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙𝑡𝑡𝑡𝑡dass 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3dass 𝜔𝜔𝜔𝜔Dann ϕdie ∙ 𝛼𝛼𝛼𝛼 3= 2 3 1 die 𝛼𝛼𝛼𝛼∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 zulässigen wird die Positionierzeit so 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 zulässigenWinkelbeschleunigung Winkelbeschleunigung führt. wird Positionierzeit so die 8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 𝑇𝑇𝑇𝑇 3 ∙ 𝑡𝑡𝑡𝑡 = − 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝛼𝛼𝛼𝛼 2 3 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 ϕ = ∙ 𝑡𝑡𝑡𝑡 − ∙ ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 2 𝑇𝑇𝑇𝑇 3 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔 = = 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ϕ= 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 − ∙ ∙ 𝑡𝑡𝑡𝑡 erlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 1 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 = ϕ ∙ 𝛼𝛼𝛼𝛼 8 2 𝑇𝑇𝑇𝑇 3 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 33 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ge nicht 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 igeWinkelbeschleunigung Winkelbeschleunigung nichtüberschreitet. überschreitet. 2 𝑇𝑇𝑇𝑇 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 𝛼𝛼𝛼𝛼 1 𝛼𝛼𝛼𝛼 Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 1 1 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 1 𝛼𝛼𝛼𝛼 8𝑠𝑠𝑠𝑠gewünschte 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3 ∙33𝑡𝑡𝑡𝑡 � ∙ 𝑡𝑡𝑡𝑡 ϕ = − ∙ 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙∙ϕ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ Position 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � hen ischen Bewegungen Bewegungen zur zur Anwendung, Anwendung, wenn wenn die die gewünschte Position innerhalb innerhalb der der geforderten geforderten 2 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 2 2 2 3 = 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔= 3𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 83 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal ϕ= = ∙ ∙2𝑡𝑡𝑡𝑡∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−∙ − 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 − ∙ 𝑡𝑡𝑡𝑡∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∙ϕ𝑡𝑡𝑡𝑡ϕ= ϕ =innerhalb − 𝑡𝑡𝑡𝑡− ∙ ∙3∙ 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 8 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 8gewünschte Verlustoptimal nischen Bewegungen zur Anwendung, wenn die Position innerhalb der geforderten Bewegungen zur Anwendung, die Position der � 2𝑇𝑇𝑇𝑇die 3 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔führt. = ϕgewünschte ∙die 𝛼𝛼𝛼𝛼 �wird Verlustoptimal Loss-optimised ulässigen ssigen Winkelbeschleunigung Winkelbeschleunigung führt. Dann Dann wird die Positionierzeit so verlängert, so2verlängert, dass 2dass 32geforderten 𝜔𝜔𝜔𝜔wenn =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼Positionierzeit Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃die 3 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ 3 ϕ 8 83 3Positionierzeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Positionierzeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung Geschwindigkeit Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 gen Winkelbeschleunigung führt. Dann wird die so verlängert, dass die 3𝑠𝑠𝑠𝑠die ulässigen Winkelbeschleunigung führt. Dann wird so verlängert, dass die 3𝑇𝑇𝑇𝑇die Winkelbeschleunigung eBewegungen Winkelbeschleunigung nicht nicht überschreitet. überschreitet. 𝜔𝜔𝜔𝜔 =� ϕPosition ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇gewünschte = stoptimal Verlustoptimal =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 newegungen Anwendung, wenn innerhalb geforderten ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = ∙ 𝛼𝛼𝛼𝛼 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 � ��ϕ�2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠ϕ 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 =∙ = ∙𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =� ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼= 8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 zurzur Anwendung, wenn die gewünschte Position innerhalb derder geforderten 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 nkelbeschleunigung nicht nicht überschreitet. e Winkelbeschleunigung überschreitet. 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 8 �2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 = �2𝜋𝜋𝜋𝜋 ϕ Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 8 8 8 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇dass 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 gen Winkelbeschleunigung führt. Dann wird Positionierzeit verlängert, die2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 die = �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 so𝑠𝑠𝑠𝑠so Verlustoptimal 𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡dass − 𝑇𝑇𝑇𝑇die 𝑠𝑠𝑠𝑠 � ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� = Winkelbeschleunigung führt. Dann wird die Positionierzeit verlängert, Verlustoptimal Verlustoptimal ptimal 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 =2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 Geschwindigkeit Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇Beschleunigung 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 kelbeschleunigung nicht überschreitet. ∙�𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �� ϕ =Beschleunigung 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Translation, Vorgabewerte: , amax ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 ϕ𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 = ∙𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Translation, Vorgabewerte: ssPos beschleunigung nicht überschreitet. Pos ,𝑇𝑇𝑇𝑇amax= 2 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡�� 𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �2𝜋𝜋𝜋𝜋 ∙ �𝑡𝑡𝑡𝑡 − Beschleunigung 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡 �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 =𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎∙= = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung Geschwindigkeit 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃sPos = Translation, Vorgabewerte: amax 2𝜋𝜋𝜋𝜋 �2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � �1 ϕ = 𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣,𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎2𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 ϕ Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 bei translatorischen 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese Bewegungsgesetze kommen Bewegungen zur die Position innerhalb 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � wenn ϕ = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕϕ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2�𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, Anwendung, wenn die gewünschte gewünschte𝜔𝜔𝜔𝜔 Position innerhalb 𝑇𝑇𝑇𝑇� =∙ ϕ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �2𝜋𝜋𝜋𝜋 22 𝜔𝜔𝜔𝜔ϕ =22= ∙𝑎𝑎𝑎𝑎 𝛼𝛼𝛼𝛼 𝑣𝑣𝑣𝑣Diese = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠� 𝑎𝑎𝑎𝑎einer 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhal 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 Anwendung Geschwindigkeit Beschleunigung 𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡 ϕϕ= 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 Positionierzeit zu Überschreitung der zulässigen Beschleunigung führt. Dann wird die Positionierzeit so verlängert, dass 𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 = 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋ϕ 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇= 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 == 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙einer 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 �2𝜋𝜋𝜋𝜋 �2𝜋𝜋𝜋𝜋 Anwendung Sinus2Sinus Positionierzeit Überschreitung der zulässigen Beschleunigung Dann die Positionierzeit so dass 𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 führt. 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Geschwindigkeit Beschleunigung Sinus𝐼𝐼𝐼𝐼 14 Sine function � wird ϕ = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� =verlängert, 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Anwendung 𝜔𝜔𝜔𝜔 = 𝜋𝜋𝜋𝜋�𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ϕzu ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung führt. wird die Positionierzeit so verlängert, das∙ Bewegungsgesetze der Translation mit Beschleunigung die nicht überschreitet. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung nus Sinus 2Tabelle 𝑎𝑎𝑎𝑎Begrenzung 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 ∙−�𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇− 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠Dann ϕder = Beschleunigung 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 Beschleunigung die𝑎𝑎𝑎𝑎zulässige zulässige Beschleunigung nicht überschreitet. 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 Beschleunigung die zulässige Beschleunigung nicht überschreitet. 𝜔𝜔𝜔𝜔Translation ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � mit Tabelle 14 Bewegungsgesetze der Begrenzung der Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Tabelle der Translation Begrenzung der Beschleunigung − � ϕ==𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝜋𝜋𝜋𝜋 mit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Bewegungsgesetze 2 14 Bewegungsgesetze Tabelle 17 der Rotation mit Begrenzung der Winkelbeschleunigung Parameter Geschwindigkeit Motion law Angular speed ∙�𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡 − ϕ= 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔=Besch =∙𝛼𝛼𝛼𝛼�1 �𝜔𝜔𝜔𝜔 ϕ=𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡Bewegungsgesetz ϕ= = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼= ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ϕ ∙ 𝑡𝑡𝑡𝑡Weg 𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝛼𝛼𝛼𝛼2𝜋𝜋𝜋𝜋 𝛼𝛼𝛼𝛼�𝑡𝑡𝑡𝑡 = 𝛼𝛼𝛼𝛼 2 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 Bewegungsgesetz Parameter Geschwindigkeit Besch 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙ 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼ϕ𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙acceleration − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠��𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡�� ϕWinkelbeschleunigung ϕ= 𝛼𝛼𝛼𝛼Angular 𝑡𝑡𝑡𝑡�� = 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚 ∙𝛼𝛼𝛼𝛼�𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠 �− = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼Weg 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Begrenzung Geschwindigkeit 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2Bewegungsgesetz belle der= Rotation der Winkelbeschleunigung �Parameter 𝜋𝜋𝜋𝜋∙mit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Tabelle Bewegungsgesetze der Rotation mit 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 21𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 Besch ∙ 𝑡𝑡𝑡𝑡217 ϕ =Sinus 𝜔𝜔𝜔𝜔 = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼Weg der 𝛼𝛼𝛼𝛼 2𝜋𝜋𝜋𝜋 = 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 ϕ17 =2Bewegungsgesetze =Begrenzung 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 2 2 Tabelle 17 Bewegungsgesetze der Rotation mit Begrenzung der Winkelbeschleunigung 2 � ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃∙ 𝛼𝛼𝛼𝛼∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2=� 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝜔𝜔𝜔𝜔𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 21 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =� ϕ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 𝑃𝑃𝑃𝑃𝛼𝛼𝛼𝛼 21 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 2� � 2 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � =+Tabelle 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼mit ∙𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = = ϕ ϕ𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕRotation ∙= 𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝛼𝛼𝛼𝛼 = ϕ −−𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡Winkelbeschleunigung + ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ ∙𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼= =−𝛼𝛼𝛼𝛼 −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼−− 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � �Sinus �ϕ2Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 Bewegungsgesetze der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙2∙𝑡𝑡𝑡𝑡17 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡Begrenzung 𝛼𝛼𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 = 𝑇𝑇𝑇𝑇= =𝜋𝜋𝜋𝜋 25 𝑎𝑎𝑎𝑎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 � 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus Sinus 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 22 𝑎𝑎𝑎𝑎 I 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 172 Bewegungsgesetze der Rotation mit Begrenzung ∙ 𝑡𝑡𝑡𝑡 I der Winkelbeschleunigung 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 I 𝑠𝑠𝑠𝑠 = 22 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 25 25 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 mit Begrenzung der Winkelbeschleunigung 2 𝑇𝑇𝑇𝑇 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 �4 Tabelle 17 Bewegungsgesetze der Rotation 25 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle 17 der mit Begrenzung der Winkelbeschleunigung 𝜔𝜔𝜔𝜔 =Rotation 𝜔𝜔𝜔𝜔𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇Rotation = ϕ=� ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙𝑎𝑎𝑎𝑎mit 𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼 − 𝛼𝛼𝛼𝛼−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼Winkelbeschleunigung ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ der 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 =𝛼𝛼𝛼𝛼−𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 ϕ� ϕ ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙17 𝛼𝛼𝛼𝛼Bewegungsgesetze ∙ 𝑡𝑡𝑡𝑡Bewegungsgesetze ∙ 𝑡𝑡𝑡𝑡− − 𝛼𝛼𝛼𝛼der ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 der �4 � 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �+ 𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle abelle Bewegungsgesetze der Rotation mit Begrenzung Begrenzung der Winkelbeschleunigung Winkelbeschleunigung Bewegungsgesetze Rotation mit Begrenzung der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃17 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔ϕ= ∙ 𝛼𝛼𝛼𝛼−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝛼𝛼𝛼𝛼25𝑎𝑎𝑎𝑎= −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + ∙ 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −2 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝜔𝜔𝜔𝜔 = � �∙ϕ𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �∙ϕ𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼�𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 III 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 22 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Time-optimised 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − Geschwindigkeitsoptimal 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III = 𝑠𝑠𝑠𝑠2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + + �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 25 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 22 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝛼𝛼𝛼𝛼 III 𝑠𝑠𝑠𝑠 = + ∙ 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 − 𝑣𝑣𝑣𝑣 = ∙ 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣 = �𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 �𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 = ϕ ∙ 𝛼𝛼𝛼𝛼 − 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 ∙ 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 � 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 25𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 = �ϕ𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −1 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 1 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 25 𝜔𝜔𝜔𝜔=𝜔𝜔𝜔𝜔�6 ==𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼 ∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡22 22 ∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 25− 25 25𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼==𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 − ∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡22−− ∙ ∙ 2 ∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡33 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ ∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 33 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �6 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 1 𝑎𝑎𝑎𝑎 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 𝑠𝑠𝑠𝑠 =2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚3 3 1 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 33 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 21 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 22 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 22 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 1𝛼𝛼𝛼𝛼 𝑎𝑎𝑎𝑎𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎 = = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 2 − 3= ∙ 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 =𝜔𝜔𝜔𝜔𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡== 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼𝛼𝛼∙∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2− 2 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 == 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 ∙∙ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2𝛼𝛼𝛼𝛼 3− 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 ∙− 𝑡𝑡𝑡𝑡 − ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 1 𝛼𝛼𝛼𝛼 1 𝑎𝑎𝑎𝑎 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 3 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 33 𝛼𝛼𝛼𝛼 2 𝑇𝑇𝑇𝑇 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣 = =𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 − 2 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡22 −∙ 𝑡𝑡𝑡𝑡 2∙3− 3𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ ∙𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑡𝑡𝑡𝑡 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 3𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = � 2 13 𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇 𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �88 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Verlustoptimal 8 ∙𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Loss-optimised 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝛼𝛼𝛼𝛼2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡−2 1−∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡 3 2∙ 𝑡𝑡𝑡𝑡 3∙Verlustoptimal Verlustoptimal 𝜔𝜔𝜔𝜔 = ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = − 2 ∙ 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 3 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 3 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 ∙ �𝑡𝑡𝑡𝑡 − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =� 𝑎𝑎𝑎𝑎� 𝑡𝑡𝑡𝑡��∙ ∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙�𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡−− 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�� ∙ ∙�1 𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔==𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡�� 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠=�𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡�� 1−−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�� 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 === 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡� 𝑇𝑇𝑇𝑇2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 ∙ �𝑡𝑡𝑡𝑡 − 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜋𝜋𝜋𝜋 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡�� 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 22 � 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑠𝑠𝑠𝑠 ∙ 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 � = 𝜋𝜋𝜋𝜋 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Sinus 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑇𝑇𝑇𝑇 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = �𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Sinus ∙ �𝑡𝑡𝑡𝑡 ∙− − 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠Sine 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 �function ∙ �1∙ − �1𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡��𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 =𝜔𝜔𝜔𝜔𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��𝑡𝑡𝑡𝑡�� 𝛼𝛼𝛼𝛼 =𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡� 𝑇𝑇𝑇𝑇�𝑡𝑡𝑡𝑡 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡� Tabelle 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡 −∙ �𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠Tabelle � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1 � 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = der 𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 �𝑡𝑡𝑡𝑡�� ∙ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �1of−angular �𝑡𝑡𝑡𝑡�� 𝑡𝑡𝑡𝑡��laws of𝜔𝜔𝜔𝜔rotation =with 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡��𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼𝛼𝛼∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡� Bewegungsgesetze Translation mit−Begrenzung der Beschleunigung Table 1714 Motion limitation acceleration Tabelle 14 Bewegungsgesetze der Translation mit Begrenzung der Beschleunigung 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑡𝑡𝑡𝑡�� 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡beschleunigung − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝜋𝜋𝜋𝜋 � 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡� elbeschleunigung − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ �1∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � �𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡��𝑡𝑡𝑡𝑡�� 𝜔𝜔𝜔𝜔 = 𝑡𝑡𝑡𝑡�� 𝛼𝛼𝛼𝛼 = 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 21 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 21

schleunigung beschleunigung 30 hleunigung beschleunigung 25 25

𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 = = 𝑎𝑎𝑎𝑎 =

Description of the load cycle

These motion laws are used in rotating movements if the desired position leads to an exceeding otation, Vorgabewerte: ϕPos , αmax , ωmax

bewerte: , α, max ,ϕωPos Rotation, Vorgabewerte: ϕϕPos ααacceleration ,, ω max max max of,ϕ the permissible angular Rotation, Vorgabewerte: ωmax max max and the permissible angular speed within the required rte: ϕVorgabewerte: αPos ωmax Pos max ion, ,Posα,,max ,ω

Segment

SegSegSeg-ment ment ment

Segment Segment

Segment SegSegSegment Seg-SegSegSegment Segment mentment ment ment ment

Rotation, Vorgabewerte: αDiese , ωmaxspeed segment Bewegungsgesetze kommen beiinserted rotatorischen Bewegungen zur Anwendung, wenn Pos , Diese max positioning time. Aϕconstant angular is then in the Bewegungen middle of thezur movement kommen bei rotatorischen Anwendung, wenn di otation, Rotation, Vorgabewerte: ϕBewegungsgesetze ,Diese ωmax Pos , αmax Vorgabewerte: ϕPositionierzeit , αBewegungsgesetze Pos max , ω max Bewegungsgesetze kommen bei rotatorischen Bewegungen Anwendung, wenn die Diese kommen bei rotatorischen Bewegungen zur zur Anwendung, wenn diegew g zubei einer Überschreitung der zulässigen Winkelbeschleunigung undgewün der zu Rotation, Vorgabewerte: ϕ , α , ω Diese Bewegungsgesetze kommen rotatorischen Bewegungen zur Anwendung, wenn die Pos max max Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung und der zulä nwendung sequence and the positioning time is extended so that the angular acceleration does not exceed Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung und der zulässige Rotation, Vorgabewerte: ϕ , einer α , Überschreitung ω Pos max max Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn Positionierzeit zu der zulässigen Winkelbeschleunigung und der zulässig Anwendung Mitte des Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit s Anwendung Positionierzeit zu einer Überschreitung der zulässigen und der zulässigen Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur Anwendung, wenn die gewünscht Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung un Diese Bewegungsgesetze kommen beiWinkelbeschleunigung rotatorischen Bewegungen zur Anwendung, wenn die Mitte des Bewegungsablaufs Bewegungsablaufs eine Konstantfahrtphase eingefügt die Positionierzeit soW Mitte eine Konstantfahrtphase eingefügt undund die Positionierzeit soder verlä angular acceleration andzu the angular speed doesder not exceed the permissible anguPositionierzeit einer Überschreitung zulässigen Winkelbeschleunigung und z ndung the permissible Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur wenn die gew Mitte des Bewegungsablaufs eine Konstantfahrtphase eingefügt und dieAnwendung, Positionierzeit sozuläss ver Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigk Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung und der zulässigen Winke Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung und der Anwendung Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigke Mitte des Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit so verlänge Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigkeit nic Mitte des eineKonstantfahrtphase Konstantfahrtphase eingefügt und die Positioni Diese Bewegungsgesetze kommen bei rotatorischen Bewegungen zur und Anwendung, wenn die gew nwendung Anwendung zu einer Überschreitung der zulässigen Winkelbeschleunigung und zulässigen Mitte desBewegungsablaufs Bewegungsablaufs eine eingefügt Positionierzeit Winkelbeschleunigung und die die zulässige ndü MittePositionierzeit des Bewegungsablaufs eineÜberschreitung Konstantfahrtphase eingefügt und dieWinkelgeschwindigkeit Positionierzeit soder verlängert, Mitte desWinkelgeschwindigkeit Bewegungsablaufs eine Konstantfahrtphase eingefügt und diedie Positionierzeit so ve lar speed. Anwendung Positionierzeit zuWinkelgeschwindigkeit einer der Winkelbeschleunigung und der zulässigen Winkelbeschleunigung die diezulässigen zulässige Winkelgeschwindigkeit Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgesch Translation, Vorgabewerte: sPos ,Winkelbeschleunigung amaxund , vmaxBewegungsablaufs Mitte des eine Konstantfahrtphase eingefügt und die Positionierzeit sonicht verlän Anwendung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindig Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigkeit nicht übers Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigkeit ewegungsgesetz Weg G Konstantfahrtphase eingefügt und die Positionierzeit so verlän Translation, Vorgabewerte: sPosParameter , a , v maxMitte maxdes Bewegungsablaufs eine Bewegungsgesetz Parameter Weg Geschw Bewegungsgesetz Parameter WegBewegungen zur Anwendung, Ge Winkelbeschleunigung und die Winkelgeschwindigkeit die zulässige Winkelgeschwindigkeit Diese Bewegungsgesetze kommen beiund translatorischen wenn die gewünschte Position innerhalbnich der g Winkelbeschleunigung die Winkelgeschwindigkeit die zulässige Winkelgeschwindigkeit nich Translation, Vorgabewerte:� sPos , amax , vmax � Dieseα Bewegungsgesetze kommen bei translatorischen Bewegungen zurund Anwendung, wennGeschwindigkeit die gewünschte führt. Position innerhalb der g tz Parameter Weg Gesc Positionierzeit zu einer Überschreitung der zulässigen Beschleunigung der zulässigen Dann wird in der Rotation, settings: Pos, Pos max , zu einer Anwendung Parameter Weg Geschwin ewegungsgesetz Parameter Weg Geschwindigk Bewegungsgesetz Parameter Weg Gesc gungsgesetz Parameter Weg Bewegungsgesetz Parameter Weg Positionierzeit Überschreitung der zulässigen Beschleunigung und dersozulässigen Geschwindigkeit führt. Dann in der Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit verlängert, dass diegewünschte Beschleunigung diewird zulässige 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚wenn Anwendung Diese Bewegungsgesetze kommen bei translatorischen die Position innerhalbB 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Rotation, 𝜔𝜔𝜔𝜔 Vorgabewerte: ϕPos , Weg α 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 max , ωmax Bewegungen zur Anwendung, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 Bewegungsgesetz Parameter Geschw 𝛼𝛼𝛼𝛼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Seg2 Bewegungsablaufs eine Konstantfahrtphase eingefügt und die Positionierzeit so verlängert, dass die Beschleunigung die zulässige B 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 und die Geschwindigkeit die zulässige Geschwindigkeit nicht überschreiten. Bewegungsgesetz Weg Geschw 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 == == =𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝜑𝜑𝜑𝜑 und = ∙ 𝑡𝑡𝑡𝑡 ∙ 𝑡𝑡𝑡𝑡Geschwindigkeit 𝜑𝜑𝜑𝜑 = = zulässigen I II der zulässigen 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼zu 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇Parameter 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Positionierzeit Überschreitung Beschleunigung der führt. Dann wird in Motion law Parameter Angle =einer 𝜑𝜑𝜑𝜑 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Geschwindigkeit 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 und die die zulässige Geschwindigkeit nicht überschreiten. Anwendung ment 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼Bewegungen 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Diese Bewegungsgesetze kommen bei2 rotatorischen zur Anwendung, die Bewegungsablaufs Konstantfahrtphase eingefügt und die Positionierzeit so𝛼𝛼𝛼𝛼verlängert, dass die Beschleunigung diewenn zuläss 2 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eine 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝛼𝛼𝛼𝛼 2 𝜔𝜔𝜔𝜔 Positionierzeit zu einer Überschreitung der zulässigen Winkelbeschleunigung und der Bewegungsgesetz Parameter Weg Geschwindigkeit Beschleunigun 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜔𝜔𝜔𝜔die zulässige 𝛼𝛼𝛼𝛼∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 eingefügt und die Geschwindigkeit Geschwindigkeit nicht überschreiten. = 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 ∙𝜑𝜑𝜑𝜑𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝜔𝜔𝜔𝜔 =zulä 𝛼𝛼𝛼𝛼 I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Anwendung 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 ==𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝐼𝐼𝐼𝐼Parameter 𝐼𝐼𝐼𝐼 = = 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = ∙2𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼== 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇 2und die Positionierzeit 𝜑𝜑𝜑𝜑 I 𝐼𝐼𝐼𝐼Geschwindigkeit Bewegungsgesetz Weg Beschleunigun 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 Mitte des Bewegungsablaufs eine Konstantfahrtphase so v 22 2=2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼==𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 2 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇 = 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑 = ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 𝜑𝜑𝜑𝜑 ∙ 𝑡𝑡𝑡𝑡 I I 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝜑𝜑𝜑𝜑 = ∙ 𝑡𝑡𝑡𝑡 I II I Winkelbeschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼==𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝜑𝜑𝜑𝜑𝐼𝐼𝐼𝐼 Winkelgeschwindigkeit 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼die∙zulässige 𝜑𝜑𝜑𝜑 und = + 𝜔𝜔𝜔𝜔∙𝐼𝐼𝐼𝐼2+ die Winkelgeschwindigkei 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Parameter IIII 𝛼𝛼𝛼𝛼− 𝜑𝜑𝜑𝜑2= = 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣= −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 ∙ 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz Weg Geschwindigkeit Beschleun 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝜑𝜑𝜑𝜑 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠 = 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2 2= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜑𝜑𝜑𝜑 =− 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇Bewegungsgesetz = − = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎 II 𝜔𝜔𝜔𝜔𝑎𝑎𝑎𝑎 Ge = 𝜑𝜑𝜑𝜑 =𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 + 𝜔𝜔𝜔𝜔2𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =𝜔𝜔𝜔𝜔 + 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Weg 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝜔𝜔𝜔𝜔2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II II Parameter 𝜔𝜔𝜔𝜔 2 ∙ 𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 = − 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 = − 2 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝑇𝑇𝑇𝑇 = − II 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 + 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 II II 𝑣𝑣𝑣𝑣𝜑𝜑𝜑𝜑 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 2 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 III 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠 = ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = − I = 𝜑𝜑𝜑𝜑 − + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 = = II 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 =+2𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 +𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠= 𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝛼𝛼𝛼𝛼 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝜔𝜔𝜔𝜔 III = Geschwindigkeitsoptimal + 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼I∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2− 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝑠𝑠𝑠𝑠 = 2𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼∙𝑚𝑚𝑚𝑚=𝑎𝑎𝑎𝑎𝜑𝜑𝜑𝜑 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 22 𝜑𝜑𝜑𝜑 + ∙ 𝑡𝑡𝑡𝑡∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 2 𝑣𝑣𝑣𝑣𝜑𝜑𝜑𝜑∙= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2− 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 2𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ +𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚− 𝛼𝛼𝛼𝛼 𝜑𝜑𝜑𝜑+ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑣𝑣𝑣𝑣𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝜔𝜔𝜔𝜔𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝛼𝛼𝛼𝛼 2 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 2 2 ∙ 𝛼𝛼𝛼𝛼 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇 + 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙2𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 = −𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 III𝑣𝑣𝑣𝑣 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − +− 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣= 𝜔𝜔𝜔𝜔+𝜔𝜔𝜔𝜔III 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜑𝜑𝜑𝜑− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝛼𝛼𝛼𝛼𝑎𝑎𝑎𝑎= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eschwindigkeitsoptimal +∙∙𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣 ∙𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 2 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝜑𝜑𝜑𝜑 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 22 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 2 𝜔𝜔𝜔𝜔 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡2 − ∙ 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼2 𝑎𝑎𝑎𝑎 = 𝑇𝑇𝑇𝑇 = +II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑠𝑠𝑠𝑠 = − 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝜑𝜑𝜑𝜑 +𝑎𝑎𝑎𝑎𝜑𝜑𝜑𝜑 𝑣𝑣𝑣𝑣= 𝑡𝑡𝑡𝑡− −− ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 22 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 III 𝜔𝜔𝜔𝜔 =− 22 𝑇𝑇𝑇𝑇 = +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔+ − 𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑃𝑃𝑃𝑃= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑣𝑣𝑣𝑣∙𝜔𝜔𝜔𝜔 III𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃III = +gewünschte ∙𝜑𝜑𝜑𝜑 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡=∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −𝛼𝛼𝛼𝛼 Time-optimised 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜑𝜑𝜑𝜑 − ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 tatorischen Bewegungen zur Anwendung, wenn die Position innerhalb geforderten Geschwindigkeitsoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑++𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 22der = − 2 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II∙𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 =𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 ∙𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 =𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑣𝑣𝑣𝑣𝜑𝜑𝜑𝜑 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ ∙𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − III 𝑇𝑇𝑇𝑇 = + ∙ 2 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠 = = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 2 ∙𝑇𝑇𝑇𝑇𝑎𝑎𝑎𝑎 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼geforderten 𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2 ∙ 𝛼𝛼𝛼𝛼 2 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝜑𝜑𝜑𝜑 − + ∙ 𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2 ∙ 𝛼𝛼𝛼𝛼 ewegungen zur Anwendung, wenn die gewünschte Position innerhalb der Geschwindigkeitsoptimal 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 2−∙∙ 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 + 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2− ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑇𝑇𝑇𝑇Winkelbeschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚− 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III zulässigen 𝜔𝜔𝜔𝜔 = 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =Winkelgeschwindigkeit 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃durch −2𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 III 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜔𝜔𝜔𝜔+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑∙ = 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔führt. 2 2 der zulässigen und Dann wird in der soptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼+ 𝑠𝑠𝑠𝑠der 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣die 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Bewegungsablauf Wird der maximal zulässige Winkelgeschwindigkeit bestimmt, ∙hat 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2in 𝛼𝛼𝛼𝛼III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 imal 𝜔𝜔𝜔𝜔Dann 2𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +Winkelgeschwindigkeit hwindigkeitsoptimal nantfahrtphase Winkelbeschleunigung und der zulässigen = 𝑣𝑣𝑣𝑣2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝑎𝑎𝑎𝑎 𝛼𝛼𝛼𝛼der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −2 ∙ 𝛼𝛼𝛼𝛼führt. + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − wird ∙ 𝑡𝑡𝑡𝑡2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚wenn 𝑎𝑎𝑎𝑎so 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎Winkelbeschleunigung 2 geforderten III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Bewegungsablauf 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eingefügt undzur dieAnwendung, Positionierzeit verlängert, dass die die 𝜑𝜑𝜑𝜑Bewegungsgesetz. = 𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −zulässige + 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Daher ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − sollte ∙ 𝑡𝑡𝑡𝑡bestimm Wird der Bewegungsablauf durch zulässige Winkelgeschwindigkeit bestimmt Wird der durch die maximal zulässige Winkelgeschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 in dies ei rotatorischen Bewegungen die gewünschte Position innerhalb der Vorteil mehr gegenüber dem geschwindigkeitsoptimalen 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeitsoptimal 2 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2hat Wird der Bewegungsablauf durch maximal zulässige bestimmt, das verl eingefügt unddie diezulässige Positionierzeit so verlängert, dass die Winkelbeschleunigung die zulässige If the motion sequence isdie determined by the Winkelgeschwindigkeit maximum Wird der Bewegungsablauf durch dieGeschwindigkeit maximal zulässige Winkelgeschwindigkeit bestimmt, h en bei rotatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten Wird der Bewegungsablauf durch die maximal zulässige bestimmt, hatpermissible das verlustoptimale Bewegungsgesetz kein eschwindigkeit Winkelgeschwindigkeit nicht überschreiten. Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in ng der zulässigen Winkelbeschleunigung und der zulässigen Winkelgeschwindigkeit führt. Dann wird in Bewegungsgesetz zurück gegriffen werden. otatorischen Bewegungen zur Anwendung, wenn die gewünschte Position innerhalb der geforderten Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte Wird der Bewegungsablauf durch die maximal zulässige Winkelgeschwindigkeit bestimmt, hat Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, hat dasder verlustoptimale Bewegungsgesetz kein mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in diesem Fall Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher solltehat in Wird der Bewegungsablauf durch die maximal zulässige Winkelgeschwindigkeit bestimmt, dd ei rotatorischen Bewegungen zurVorteil Anwendung, wenn die gewünschte Position innerhalb der geforderten angular speed, the loss-optimised law has no advantage over the reitung der zulässigen Winkelbeschleunigung und der Winkelgeschwindigkeit führt. Dann wird in der tder die zulässige Winkelgeschwindigkeit nicht überschreiten. mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte Bewegungsgesetz. in diesem nstantfahrtphase eingefügt und die Positionierzeit so zulässigen verlängert, dass die Winkelbeschleunigung die zulässige Wird der Bewegungsablauf durch diemotion maximal zulässige Winkelgeschwindigkeit bestim Bewegungsgesetz zurück gegriffen zulässigen Winkelbeschleunigung und dergewünschte zulässigen Winkelgeschwindigkeit führt. Dann wird inFall derauf das geschwindigkeitsoptimale Bewegungsgesetz zurück gegriffen werden. Bewegungsgesetz zurück gegriffen werden. Vorteil gegenüber dem geschwindigkeitsoptimalen Dahersollte solltein indiesem diese orischen Bewegungen zur Anwendung, wenn die Position innerhalb der geforderten Bewegungsgesetz zurück gegriffen werden. Bewegungsgesetz zurück gegriffen werden. Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher ung der zulässigen Winkelbeschleunigung und dermehr zulässigen Winkelgeschwindigkeit führt. Dann wird indas der e Konstantfahrtphase eingefügt und die Positionierzeit so verlängert, dass die diehat zulässige Wird der Winkelbeschleunigung Bewegungsablauf durch die maximal zulässige Winkelgeschwindigkeit bestimmt, time-optimised motion law. In this case, the time-optimised motion law Bewegungsgesetz zurück gegriffen werden. Verlustoptimal Wird der Bewegungsablauf durch die maximal zulässige Geschwindigkeit bestimmt, verlustoptimale Bewegungsgesetz Wird derdie Bewegungsablauf durch diedass maximal zulässige Winkelgeschwindigkeit bestimmt, ha elgeschwindigkeit die zulässige Winkelgeschwindigkeit nichtzurück überschreiten. antfahrtphase eingefügt und Positionierzeit so verlängert, die Winkelbeschleunigung dieinBewegungsgesetz. zulässige Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Daher sollte Bewegungsgesetz gegriffen werden. zulässigen Winkelbeschleunigung und der zulässigen Winkelgeschwindigkeit führt. Dann wird in der Geschwindigkeit Beschleunigung Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Daher in Wird der durch die maximal zulässige Winkelgeschwindigkeit bestimmt, hatsollte das mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher solltedie diesem Fall Bewegungsgesetz. auf das geschwindigkeitsoptim Wird der Bewegungsablauf durch die zulässige Winkelgeschwindigkeit Bewegungsgesetz zurück gegriffen werden. Winkelgeschwindigkeit dieBewegungsablauf zulässige Winkelgeschwindigkeit nicht überschreiten. nstantfahrtphase eingefügt und die Positionierzeit so verlängert, dass die Winkelbeschleunigung zulässige Verlustoptimal should be used. Verlustoptimal ungen zur Anwendung, wenn die gewünschte Position innerhalb der maximal geforderten Bewegungsgesetz zurückdie gegriffen werden. erlustoptimal geschwindigkeit die Vorteil zulässige Winkelgeschwindigkeit nicht überschreiten. Geschwindigkeit Beschleunigung erlustoptimal ahrtphase eingefügt und die Positionierzeit so verlängert, dass die Winkelbeschleunigung zulässige Verlustoptimal Bewegungsgesetz zurück gegriffen werden. mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in die Loss-optimised Bewegungsgesetz zurück gegriffen werden. 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 elgeschwindigkeit die zulässige Winkelgeschwindigkeit nicht überschreiten. 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚geschwindigkeitsoptimalen 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz. 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Vorteil mehr gegenüber dem geschwindigkeitsoptimalen Bewegungsgesetz. Daher sollte in diesem 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚gegenüber dem Vorteil mehr Dahe Verlustoptimal 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = ∙ �1 − I 𝑠𝑠𝑠𝑠 = 𝑣𝑣𝑣𝑣 ∙ �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼führt. − 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠Dann 𝑠𝑠𝑠𝑠 �2𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 ∙ �1 −𝐼𝐼𝐼𝐼𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐 I kelbeschleunigung und der zulässigen Winkelgeschwindigkeit wird in der 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Verlustoptimal windigkeit die zulässige Winkelgeschwindigkeit nicht überschreiten. 𝜋𝜋𝜋𝜋 g Geschwindigkeit Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz zurück gegriffen 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼Beschleunigung 𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Beschleunigung 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 �� 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 2𝜔𝜔𝜔𝜔𝑎𝑎𝑎𝑎𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 =𝜋𝜋𝜋𝜋2 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 gegriffen = 2 ∙ �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 − 2𝑎𝑎𝑎𝑎𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 = 𝜔𝜔𝜔𝜔2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑣𝑣𝑣𝑣𝜔𝜔𝜔𝜔 =𝜔𝜔𝜔𝜔 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋Izurück 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚werden. 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋2𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔�𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Weg Geschwindigkeit 𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼gegriffen 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz zurück werden. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal werden. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Translation, Vorgabewerte: sBewegungsgesetz , 𝑇𝑇𝑇𝑇a𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ,𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃v𝑇𝑇𝑇𝑇 2𝜋𝜋𝜋𝜋𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Pos max max 2wenn die 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎2𝜔𝜔𝜔𝜔 𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡 2� 2𝛼𝛼𝛼𝛼 Verlustoptimal 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝑇𝑇 = 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠 efügt und die Positionierzeit so verlängert, dass die Winkelbeschleunigung die zulässige 𝜑𝜑𝜑𝜑 = ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝜔𝜔𝜔𝜔 = ∙ �∙ 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇 = = 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 I ngen zur Anwendung, gewünschte Position innerhalb der geforderten = 𝑇𝑇𝑇𝑇 = 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = ∙ �1 − �∙ �1 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 = ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼= =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ Beschleunigung 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = Geschwindigkeit 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼−− 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 I I I𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠I∙𝑚𝑚𝑚𝑚=2𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝐼𝐼𝐼𝐼+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼�𝑡𝑡𝑡𝑡 �� 𝐼𝐼𝐼𝐼=𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Anwendung, 𝑇𝑇𝑇𝑇= 𝑣𝑣𝑣𝑣 =𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎∙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Beschleunigung 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 ��𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝜔𝜔𝜔𝜔 2 = �1 2 𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 I𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 g Geschwindigkeit 𝑇𝑇𝑇𝑇Position = 2𝐼𝐼𝐼𝐼==2𝜋𝜋𝜋𝜋𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼der 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 �𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝜔𝜔𝜔𝜔 2 2𝛼𝛼𝛼𝛼 2 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝐼𝐼 − 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 2II 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚2𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 ∙ − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � �� 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝐼𝐼𝐼𝐼 − 𝑚𝑚𝑚𝑚 2 𝜔𝜔𝜔𝜔 2 2𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 n zur die gewünschte innerhalb geforderten 2 wenn 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2 2𝛼𝛼𝛼𝛼 2 2 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 = 𝜑𝜑𝜑𝜑 ∙ �𝑡𝑡𝑡𝑡der −𝜔𝜔𝜔𝜔+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼 � = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝑣𝑣𝑣𝑣 𝜔𝜔𝜔𝜔 =wenn ∙�� 1 − �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= Position 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��2 ∙innerhalb 𝑎𝑎𝑎𝑎 − = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I ∙ 𝑡𝑡𝑡𝑡 innerhalb Geschwindigkeit Beschleunigung 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡= 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚gewünschte = 2 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑠𝑠𝑠𝑠 ==führt. 𝑣𝑣𝑣𝑣∙Dann 𝐼𝐼𝐼𝐼nicht 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼Winkelgeschwindigkeit 2 2𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 = zulässige überschreiten. gen zur ∙Anwendung, wenn die gewünschte Position geforderten 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 Diese Bewegungsgesetze kommen bei translatorischen Bewegungen zur Anwendung, die 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼und 𝑚𝑚𝑚𝑚�𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 lbeschleunigung der zulässigen Winkelgeschwindigkeit wird in der − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � �1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 2 2𝑎𝑎𝑎𝑎 2 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 4=𝑎𝑎𝑎𝑎geforderten 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 die gewünschte 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 II 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 2 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 en zur Anwendung, wenn Position innerhalb der 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 stoptimal 2 2 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝜑𝜑𝜑𝜑 = ∙ 𝑡𝑡𝑡𝑡 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼²𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣4 𝛼𝛼𝛼𝛼der 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋𝛼𝛼𝛼𝛼 𝜋𝜋𝜋𝜋 führt. 𝑣𝑣𝑣𝑣∙zulässigen eschleunigung und der 𝐼𝐼𝐼𝐼zulässigen wird inund Positionierzeit zu einer Überschreitung zulässigen in 2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung 2Dann 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙2 �𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =𝑠𝑠𝑠𝑠𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜑𝜑𝜑𝜑𝑇𝑇𝑇𝑇Winkelgeschwindigkeit 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �∙ Geschwindigkeit 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� führt.𝜔𝜔𝜔𝜔Dann =diewird 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Anwendung 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 Ider 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼−∙Dann 𝜑𝜑𝜑𝜑und = ∙zulässigen 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 2so verlängert, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =−𝛼𝛼𝛼𝛼 = 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 𝑣𝑣𝑣𝑣 𝐼𝐼𝐼𝐼 Winkelgeschwindigkeit 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 die 𝜋𝜋𝜋𝜋𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 beschleunigung führt. in==𝛼𝛼𝛼𝛼der der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣die 2 der ügt und=die Positionierzeit dass Winkelbeschleunigung die 2 wird 𝐼𝐼𝐼𝐼II− zulässige 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼und = 𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ²∙ +𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2Positionierzeit 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 ∙ 𝑡𝑡𝑡𝑡= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 eine Konstantfahrtphase eingefügt so𝑣𝑣𝑣𝑣= verlängert, zuläss 𝑇𝑇𝑇𝑇Winkelgeschwindigkeit − 4 2𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 die 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 Beschleunigung IIII𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 = 2𝜋𝜋𝜋𝜋 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 =Bewegungsablaufs − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔dass 2 = 𝜔𝜔𝜔𝜔 22 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 beschleunigung der führt. Dann wird in 𝜔𝜔𝜔𝜔 0𝑚𝑚𝑚𝑚2+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙∙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 +der =𝜋𝜋𝜋𝜋𝜑𝜑𝜑𝜑𝑣𝑣𝑣𝑣𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑠𝑠𝑠𝑠= == 𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼 − + ∙𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼verlängert, 𝜔𝜔𝜔𝜔 =𝑎𝑎𝑎𝑎 =𝜔𝜔𝜔𝜔𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4∙𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔 = +und 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔 = 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑 = 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜑𝜑𝜑𝜑 und diedie Positionierzeit dass die Winkelbeschleunigung die zulässige 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 II = ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2∙2𝑡𝑡𝑡𝑡zulässigen 𝜔𝜔𝜔𝜔zulässige ∙𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡∙𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝜑𝜑𝜑𝜑 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼∙ = 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔 ∙𝛼𝛼𝛼𝛼 4= 𝑎𝑎𝑎𝑎2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2so 𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 == +2𝜋𝜋𝜋𝜋2𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝜑𝜑𝜑𝜑∙Positionierzeit 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔 � 4II III 𝑇𝑇𝑇𝑇 = − 𝑇𝑇𝑇𝑇 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 und die Geschwindigkeit die Geschwindigkeit nicht überschreiten. II 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜑𝜑𝜑𝜑 = 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡 = 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 4 gt und so verlängert, dass die Winkelbeschleunigung die zulässige 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 2 4 𝛼𝛼𝛼𝛼 𝑣𝑣𝑣𝑣 2 𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 lässige Winkelgeschwindigkeit nicht überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼�𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝐼𝐼𝐼𝐼 2𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼2 =𝐼𝐼𝐼𝐼 so 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4− 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜔𝜔𝜔𝜔− 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝜑𝜑𝜑𝜑 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼verlängert, = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 =Sinus ∙ 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠− � 𝜔𝜔𝜔𝜔= 𝛼𝛼𝛼𝛼 ∙= 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼II 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2I=Winkelbeschleunigung 𝛼𝛼𝛼𝛼 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝜔𝜔𝜔𝜔��𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙= �1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 gtsige undWinkelgeschwindigkeit Positionierzeit dass die die zulässige 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = = 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 0 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +die ∙ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝜑𝜑𝜑𝜑 𝐼𝐼𝐼𝐼 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+ �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼= 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋 − 𝑣𝑣𝑣𝑣= ²−𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣4 𝑣𝑣𝑣𝑣4 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚 2= 𝑎𝑎𝑎𝑎 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ��𝑡𝑡𝑡𝑡 ∙�� 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ nicht überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 = �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝜑𝜑𝜑𝜑 = ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � I 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 0 𝜑𝜑𝜑𝜑 = ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝜔𝜔𝜔𝜔 = 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 I 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼sPos2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 �� 𝐼𝐼𝐼𝐼 2 Translation, Vorgabewerte: , a𝛼𝛼𝛼𝛼 , 𝜔𝜔𝜔𝜔 vmax Sinus 𝑣𝑣𝑣𝑣 0 = 𝐼𝐼𝐼𝐼max 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 �𝑡𝑡𝑡𝑡 4−+2𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 =𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −𝜔𝜔𝜔𝜔 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 2 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ässige Winkelgeschwindigkeit nicht überschreiten. �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝛼𝛼𝛼𝛼 2 𝛼𝛼𝛼𝛼 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 = 𝛼𝛼𝛼𝛼 = 2 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 2 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Winkelgeschwindigkeit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼nicht 4𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz Geschwindigkeit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Parameter 2 ∙ 𝛼𝛼𝛼𝛼2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜋𝜋𝜋𝜋𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 2 𝜔𝜔𝜔𝜔𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣− ässige überschreiten. 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+𝜑𝜑𝜑𝜑 = 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑+ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleun 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣+ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Weg 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑= + ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 III 𝜔𝜔𝜔𝜔 =𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1∙ = 𝜑𝜑𝜑𝜑 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =2 + ∙ 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sinus 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + == 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼= =−𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼III𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Beschleunigung ∙ 𝜔𝜔𝜔𝜔𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼+ 2𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃Sinus −𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔+ 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙ 𝑡𝑡𝑡𝑡−𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙Geschwindigkeit ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜑𝜑𝜑𝜑 = ∙ ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑 + III = ∙ �12−der 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐g 𝑣𝑣𝑣𝑣 𝜔𝜔𝜔𝜔 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 Bewegungsgesetze bei translatorischen Bewegungen zur Anwendung, die gewünschte innerhalb 2kommen 𝛼𝛼𝛼𝛼=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 + Position 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 2𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔 =02wenn 0∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ) 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑 2=∙ 𝛼𝛼𝛼𝛼++ + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 Diese 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 𝜑𝜑𝜑𝜑𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼= == 𝜑𝜑𝜑𝜑 +𝛼𝛼𝛼𝛼+��2𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔2𝜋𝜋𝜋𝜋=𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 0 �(𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2) −+Geschwindigkeit 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ��𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔+ �� 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 −) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Dann ��𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 = 22 ∙ 𝛼𝛼𝛼𝛼Sinus Positionierzeit zu einer der Beschleunigung der führt. wird in der𝐼𝐼𝐼𝐼 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − �𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +und 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼zulässigen 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜔𝜔𝜔𝜔 ∙function 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔 ∙ 2𝐼𝐼𝐼𝐼 2𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡�1 𝜔𝜔𝜔𝜔 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 2 zulässigen 𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜋𝜋𝜋𝜋Überschreitung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 2+ 2𝜋𝜋𝜋𝜋𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 Geschwindigkeit Beschleunigung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝜑𝜑𝜑𝜑 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = + 2 Anwendung 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙− 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Sine 𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝜑𝜑𝜑𝜑 = +𝜋𝜋𝜋𝜋= 2𝑎𝑎𝑎𝑎𝜑𝜑𝜑𝜑 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝜔𝜔𝜔𝜔 =𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ �1 −die 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �∙−�1𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔𝜔𝜔 III 𝜔𝜔𝜔𝜔 = − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇∙𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡= =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡22 =und ∙ 𝑡𝑡𝑡𝑡− 𝑣𝑣𝑣𝑣 = ∙𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 = 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 ∙𝜑𝜑𝜑𝜑 𝑡𝑡𝑡𝑡𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡 = 𝛼𝛼𝛼𝛼 I∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � �� 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 − 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 − 𝛼𝛼𝛼𝛼 ∙ −𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼= 𝐼𝐼𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 =+ 𝜑𝜑𝜑𝜑 + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 − 𝜋𝜋𝜋𝜋 2 𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇 = + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 2 inus 𝑇𝑇𝑇𝑇 = − Bewegungsablaufs eine Konstantfahrtphase eingefügt die Positionierzeit so verlängert, dass die Beschleunigung zulässige Geschwindigkeit Beschleunigung 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III ∙ �1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 = + II 𝜔𝜔𝜔𝜔 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝜑𝜑𝜑𝜑 = + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛼𝛼𝛼𝛼Rotation 𝛼𝛼𝛼𝛼+Winkelbeschleunigung =𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 −𝛼𝛼𝛼𝛼 2mit IIIBeschleunigung 𝜔𝜔𝜔𝜔 ∙ �1B 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 = 𝜑𝜑𝜑𝜑Sinus ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − ∙Tabelle 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜑𝜑𝜑𝜑𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋𝜑𝜑𝜑𝜑 𝜔𝜔𝜔𝜔 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚22∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 )𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 18 Bewegungsgesetze Begrenzung der der Winkelgeschwi 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 ) 2und𝜔𝜔𝜔𝜔 2 ∙−𝛼𝛼𝛼𝛼𝑇𝑇𝑇𝑇 2und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 Sinus 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = − 2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼+ 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 =2 𝜔𝜔𝜔𝜔 2= 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 = + 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡+ = +𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡überschreiten. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙� 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝜔𝜔𝜔𝜔 + ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2die 𝛼𝛼𝛼𝛼2zulässige die Geschwindigkeit Geschwindigkeit 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙2𝛼𝛼𝛼𝛼= 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4III𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼II 𝛼𝛼𝛼𝛼der 𝐼𝐼𝐼𝐼�� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 )𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 inus )𝑡𝑡𝑡𝑡𝜑𝜑𝜑𝜑 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −nicht 𝑠𝑠𝑠𝑠= �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III�(𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 =∙ �1 −𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 )𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠− 2 𝜔𝜔𝜔𝜔 �(𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼)𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 �𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙+ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 )�� 2 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 2𝐼𝐼𝐼𝐼 + 2 2 4 𝛼𝛼𝛼𝛼 �(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � �� 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 − ∙ 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑=Sinus − + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 ) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 Motion law Angular speed Angular acceleration 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2der 2und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 der ∙Winkelbeschleunigung 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇=𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2−∙ 18 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙−𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙=𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑇𝑇𝑇𝑇)𝐼𝐼𝐼𝐼 der 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣Winkelgeschwindig 2𝛼𝛼𝛼𝛼 +2𝛼𝛼𝛼𝛼𝑡𝑡𝑡𝑡+𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − �𝐼𝐼𝐼𝐼 + �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠−𝛼𝛼𝛼𝛼 Tabelle Bewegungsgesetze Begrenzung 𝜔𝜔𝜔𝜔−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜔𝜔𝜔𝜔mit −= 𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡− = −𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = II 𝑠𝑠𝑠𝑠 =�(𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣𝛼𝛼𝛼𝛼) ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡2𝛼𝛼𝛼𝛼 𝜑𝜑𝜑𝜑 𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Parameter 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 22 + − �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �� 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑎𝑎𝑎𝑎𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼 2𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Rotation 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼26�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Geschwindigkeit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼zulässige 2 �(𝑇𝑇𝑇𝑇�(𝑇𝑇𝑇𝑇 −𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼 2∙18 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡das =𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Bewegungsgesetz 𝐼𝐼𝐼𝐼 2+∙ 𝐼𝐼𝐼𝐼𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Bewegungsgesetz = + aximal Winkelgeschwindigkeit bestimmt, hat∙mit verlustoptimale keinen 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Weg 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 IIIWinkelbeschleunigung 𝜔𝜔𝜔𝜔Beschleunigun = ∙ 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼derder 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 belle Bewegungsgesetze der Rotation mit Begrenzung der und der Winkelgeschwindigkeit 2𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Tabelle 18 Bewegungsgesetze der Rotation Begrenzung der Winkelbeschleunigung und Winkelgeschwin 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 0 𝜔𝜔𝜔𝜔 Sinus + 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Tabelle 18 Bewegungsgesetze der Rotation mit Begrenzung der Winkelbeschleunigung und Winkelgeschwindigk 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼gkeitsoptimalen 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 2 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ch 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙ 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Tabelle 18 Bewegungsgesetze der Rotation mit Begrenzung der Winkelbeschleunigung und der Winkelgeschw 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 maximal 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼 hat 𝐼𝐼𝐼𝐼hat 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 e 𝐼𝐼𝐼𝐼maximal zulässige Winkelgeschwindigkeit bestimmt, dasdas verlustoptimale Bewegungsgesetz keinen 𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 2 𝑇𝑇𝑇𝑇 = + 𝑎𝑎𝑎𝑎 die zulässige Winkelgeschwindigkeit bestimmt, verlustoptimale Bewegungsgesetz keinen 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣 − 𝑎𝑎𝑎𝑎 ∙ 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 III 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 − + 𝑣𝑣𝑣𝑣 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 26 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � �(𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 − 2keinen ��Winkelgeschw 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Winkelgeschwindigk 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 belle 1818 Bewegungsgesetze der mitBegrenzung und 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃der ige Winkelgeschwindigkeit verlustoptimale Bewegungsgesetz Tabelle Bewegungsgesetze der𝜋𝜋𝜋𝜋 Rotation mit Winkelbeschleunigung und𝑣𝑣𝑣𝑣 2 der 𝐼𝐼𝐼𝐼 + 𝜑𝜑𝜑𝜑bestimmt, 𝜔𝜔𝜔𝜔hat 𝑇𝑇𝑇𝑇Rotation 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =das 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠∙ 𝑎𝑎𝑎𝑎 =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Winkelbeschleunigung ∙ 𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡der 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎 IBegrenzung Geschwindigkeitsoptimal 2der 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 en. indigkeitsoptimalen Bewegungsgesetz. Daher in 𝜋𝜋𝜋𝜋 diesem FallFall aufauf das geschwindigkeitsoptimale 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2geschwindigkeitsoptimale 𝜑𝜑𝜑𝜑Bewegungsgesetz. 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔+ 𝜑𝜑𝜑𝜑sollte 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 schwindigkeitsoptimalen Daher sollte in𝜔𝜔𝜔𝜔das diesem das 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = 26 III 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝑐𝑐𝑐𝑐𝜋𝜋𝜋𝜋 26 len Bewegungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale maximal zulässige Winkelgeschwindigkeit bestimmt, hat verlustoptimale Bewegungsgesetz keinen 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 26 e𝜔𝜔𝜔𝜔 maximal bestimmt, hat das verlustoptimale Bewegungsgesetz keinen 𝜔𝜔𝜔𝜔 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔III =das 𝜔𝜔𝜔𝜔 verlustoptimale 𝛼𝛼𝛼𝛼 keinen =0 mal zulässige Winkelgeschwindigkeit hat 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =Winkelgeschwindigkeit +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇bestimmt, ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼zulässige = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 werden. IIIBewegungsgesetz 𝜔𝜔𝜔𝜔 =� 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝛼𝛼𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 fen werden. 2 𝜔𝜔𝜔𝜔 2 𝑣𝑣𝑣𝑣der 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ) 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 26 abelle 18 Bewegungsgesetze der Rotation mit Begrenzung Winkelbeschleunigung und der Winkelgesch 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 0 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 − 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡 − ∙ 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼Daher = in − digkeitsoptimalen Bewegungsgesetz. Daher sollte inin diesem Fall 𝜔𝜔𝜔𝜔 2diesem 𝛼𝛼𝛼𝛼auf 2𝑡𝑡𝑡𝑡 ) 𝑣𝑣𝑣𝑣 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑎𝑎𝑎𝑎 = II𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠das = +𝛼𝛼𝛼𝛼 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼auf 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Daher𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 indigkeitsoptimalen Bewegungsgesetz. sollte Fall auf geschwindigkeitsoptimale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz. 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡0 𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼 itsoptimalen sollte diesem Fall das geschwindigkeitsoptimale 𝜔𝜔𝜔𝜔dasgeschwindigkeitsoptimale 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝑡𝑡𝑡𝑡 ) 𝜔𝜔𝜔𝜔 26+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝜔𝜔𝜔𝜔 =𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 )2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = 𝐼𝐼𝐼𝐼�0 26𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝐼𝐼𝐼𝐼 �(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 ) − −�(𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝑠𝑠𝑠𝑠 − �� 2𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+ 𝑇𝑇𝑇𝑇��𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 2𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 2𝛼𝛼𝛼𝛼 𝑇𝑇𝑇𝑇 𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 − 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 −𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡 ungsgesetze mit Winkelbeschleunigung und der Winkelgeschwind 𝜔𝜔𝜔𝜔Begrenzung =Rotation 𝜔𝜔𝜔𝜔Begrenzung 𝛼𝛼𝛼𝛼mit 𝑡𝑡𝑡𝑡der == −𝛼𝛼𝛼𝛼 mitder Winkelbeschleunigung ∙18 𝑡𝑡𝑡𝑡∙ 𝑡𝑡𝑡𝑡Bewegungsgesetze − − der2∙ Rotation 𝑡𝑡𝑡𝑡der Rotation egesetze der𝛼𝛼𝛼𝛼Winkelbeschleunigung und der Winkel 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 − − 𝛼𝛼𝛼𝛼der∙Begrenzung ∙ 𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 −𝛼𝛼𝛼𝛼 und der Winkelgeschwindigkei 2 2 ∙ 𝑡𝑡𝑡𝑡

𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 den. werden.

𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝐼𝐼𝐼𝐼die maximal 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 zulässige Geschwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz Wird der Bewegungsablauf 𝐼𝐼𝐼𝐼 durch𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 in diesem Fall auf das geschwindigkeitsoptim 2 Bewegungsgesetz. 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝜔𝜔𝜔𝜔 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 mehr gegenüber dem geschwindigkeitsoptimalen 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Daher 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 sollte 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 = − III 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 =werden. 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑡𝑡𝑡𝑡 �� + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = − 𝛼𝛼𝛼𝛼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = ∙ �1 − � 𝑠𝑠𝑠𝑠2𝛼𝛼𝛼𝛼 � 2𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼−�𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Bewegungsgesetz zurück gegriffen 26 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎 Geschwindigkeitsoptimal 2 ∙ 𝑎𝑎𝑎𝑎 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 2= ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔�𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 2𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Time-optimised =2 𝜑𝜑𝜑𝜑∙ = ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 � ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 � 2 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 = ∙ 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 − 𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼 2 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 2𝛼𝛼𝛼𝛼2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2 𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔�𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Verlustoptimal 𝜔𝜔𝜔𝜔 𝑡𝑡𝑡𝑡2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ��𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝛼𝛼𝛼𝛼2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔hat ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − � 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝛼𝛼𝛼𝛼 �keinen 𝑡𝑡𝑡𝑡2𝛼𝛼𝛼𝛼 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Winkelgeschwindigkeit bestimmt, das verlustoptimale Bewegungsgesetz 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑣𝑣𝑣𝑣∙𝑚𝑚𝑚𝑚�1 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡�𝐼𝐼𝐼𝐼 �� 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 ∙= �1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝜋𝜋𝜋𝜋 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙= 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼2𝑎𝑎𝑎𝑎 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �� 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∙ �𝑡𝑡𝑡𝑡∙𝐼𝐼𝐼𝐼 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 �� = −− 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼die 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 ∙bestimmt, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼2�� ∙ ��1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 ��maximal 𝑡𝑡𝑡𝑡�� 𝛼𝛼𝛼𝛼 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼� 𝐼𝐼𝐼𝐼 � 2−𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼∙��1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 ∙ 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼der = 𝑇𝑇𝑇𝑇Bewegungsablauf = 2 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= Wird durch zulässige Geschwindigkeit hat das𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 verlustoptimale Bewegungsgesetz 𝑠𝑠𝑠𝑠 = ∙ �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝑣𝑣𝑣𝑣 = 𝑡𝑡𝑡𝑡 �� 𝑎𝑎𝑎𝑎 = kein 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 I 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 2𝛼𝛼𝛼𝛼 2 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 0 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 + 𝜔𝜔𝜔𝜔 2 2𝛼𝛼𝛼𝛼 2 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋 𝜔𝜔𝜔𝜔 2 2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 geschwindigkeitsoptimale ewegungsgesetz. Daher 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 sollte inmehr diesem Fall das 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚auf 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 26 𝑣𝑣𝑣𝑣Daher 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 Bewegungsgesetz. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Fall 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 gegenüber dem auf das geschwindigkeitsoptimale 𝜔𝜔𝜔𝜔geschwindigkeitsoptimalen =𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 sollte = =𝛼𝛼𝛼𝛼𝜑𝜑𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 262𝑎𝑎𝑎𝑎2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 0=in0diesem = + 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 26 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣zurück 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 42 𝛼𝛼𝛼𝛼2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Bewegungsgesetz gegriffen werden. 24 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = − 𝑣𝑣𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 II 𝑠𝑠𝑠𝑠 = + 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔𝜋𝜋𝜋𝜋𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 bestimmt, hat das 𝜔𝜔𝜔𝜔 = ∙ 𝑡𝑡𝑡𝑡+𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔= 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ==0 0 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼==00 ∙ 𝑡𝑡𝑡𝑡𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 verlustoptimale Bewegungsgesetz keinen 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼+ 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔= =𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =𝑚𝑚𝑚𝑚elgeschwindigkeit + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 +𝜑𝜑𝜑𝜑𝜔𝜔𝜔𝜔 ∙ 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼∙𝜔𝜔𝜔𝜔 𝜑𝜑𝜑𝜑𝛼𝛼𝛼𝛼𝜔𝜔𝜔𝜔=𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Verlustoptimal 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚𝛼𝛼𝛼𝛼 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 ² 𝑣𝑣𝑣𝑣 4𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∙ ∙ in diesem Fall auf das geschwindigkeitsoptimale = Daher 𝜑𝜑𝜑𝜑+ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 + 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣 = ∙ 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − ) + ∙ gungsgesetz. 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) �2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 �� 2𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 +𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 +2∙ 𝑡𝑡𝑡𝑡�1 2 sollte 2 4 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝜋𝜋𝜋𝜋𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 = 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝜔𝜔𝜔𝜔 ) 2𝑎𝑎𝑎𝑎 2 𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=� 𝑣𝑣𝑣𝑣𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 ��∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 = − )𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 I III� 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝐼𝐼𝐼𝐼 − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 +𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡2+ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∙) 𝐼𝐼𝐼𝐼 +𝐼𝐼𝐼𝐼2 𝑡𝑡𝑡𝑡+ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡)𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 =𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃==𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃= �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �� 𝑣𝑣𝑣𝑣𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 � 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 2+ 𝑎𝑎𝑎𝑎∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � � � ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝜑𝜑𝜑𝜑 = 𝜑𝜑𝜑𝜑 + ∙ + 2𝛼𝛼𝛼𝛼 ∙ ) Sinus 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝜑𝜑𝜑𝜑𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜑𝜑𝜑𝜑 + ∙ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼 2 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 2 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋 �𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼 − 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) ) 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + +𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� Sine function 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 2 ) ) 𝑣𝑣𝑣𝑣 2 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡 𝑇𝑇𝑇𝑇 = − 2 𝑎𝑎𝑎𝑎 = II ��𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 −+) 𝑡𝑡𝑡𝑡−𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ) 𝑡𝑡𝑡𝑡−𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 == + ∙ 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠 � 𝐼𝐼𝐼𝐼𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼− 𝐼𝐼𝐼𝐼 2𝑎𝑎𝑎𝑎𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �)𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ��𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝑇𝑇𝑇𝑇�= 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔�� 2𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼=))𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �(𝑇𝑇𝑇𝑇 �� 𝑣𝑣𝑣𝑣 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + ��∙ �1 )= ) �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 � 4) 𝑎𝑎𝑎𝑎𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 + 𝐼𝐼𝐼𝐼)𝐼𝐼𝐼𝐼++𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = ∙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �1 � ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 + 𝐼𝐼𝐼𝐼𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼 +𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 2 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 −angular 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 ∙𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � � =−=𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �1 − �� 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼= �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2�1𝜔𝜔𝜔𝜔 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼� � �� 𝜔𝜔𝜔𝜔 = of𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼rotating ∙𝜔𝜔𝜔𝜔 �∙𝜔𝜔𝜔𝜔 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙= 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼𝑠𝑠𝑠𝑠𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � ∙𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )laws 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Table 18𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Motion with limitation of𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 acceleration 𝑣𝑣𝑣𝑣𝑇𝑇𝑇𝑇 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ² and 𝐼𝐼𝐼𝐼 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇 2 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 angular speed ) 2 𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + 𝑡𝑡𝑡𝑡 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 )𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼2 Winkelgeschwindigkeit 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼− 4 𝑎𝑎𝑎𝑎 + 2 ∙ 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑣𝑣𝑣𝑣 = 2𝐼𝐼𝐼𝐼 ∙ 𝜔𝜔𝜔𝜔𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼der − Winkelbeschleunigung 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ) 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 und 𝜋𝜋𝜋𝜋(𝑇𝑇𝑇𝑇 + nkelbeschleunigung und Winkelgeschwindigkeit Winkelbeschleunigung der Winkelgeschwindigkeit ) −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑡𝑡𝑡𝑡)� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�𝑇𝑇𝑇𝑇𝑠𝑠𝑠𝑠𝐼𝐼𝐼𝐼 �𝜔𝜔𝜔𝜔 ��und ��der 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡der − 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝐼𝐼𝐼𝐼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 22 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �� 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =2𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑎𝑎𝑎𝑎 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � III 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼��𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 2𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 𝜔𝜔𝜔𝜔 = � 𝑡𝑡𝑡𝑡 �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 � 𝑣𝑣𝑣𝑣 2𝑎𝑎𝑎𝑎 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = 0 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 Sinus 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 �� lbeschleunigung und der Winkelgeschwindigkeit 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

kelgeschwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz keinen geschwindigkeit bestimmt, hat das verlustoptimale Bewegungsgesetz keinen kelgeschwindigkeit bestimmt, das verlustoptimale Bewegungsgesetz keinen wegungsgesetz. Daher sollte inhat diesem Fall auf das geschwindigkeitsoptimale ungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale egungsgesetz. Daher sollte in diesem Fall auf das geschwindigkeitsoptimale

2𝛼𝛼𝛼𝛼 𝜔𝜔𝜔𝜔 2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼�2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼�2𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼�2𝛼𝛼𝛼𝛼 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 =𝜔𝜔𝜔𝜔 ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 � 𝑡𝑡𝑡𝑡 �� 𝜔𝜔𝜔𝜔 = ∙ �1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 �� 𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝜔𝜔𝜔𝜔 2 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 �𝑡𝑡𝑡𝑡 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝜋𝜋𝜋𝜋 + 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡�� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼𝐼𝐼 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 �� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝐼𝐼𝐼𝐼 𝑡𝑡𝑡𝑡 �� 2und 𝜔𝜔𝜔𝜔𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2𝑎𝑎𝑎𝑎 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 Winkelbeschleunigung und Winkelgeschwindigkeit 𝜔𝜔𝜔𝜔26= 2 ∙ der �1Winkelgeschwindigkeit − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 �𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝐼𝐼𝐼𝐼 � 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 inkelbeschleunigung der 𝐼𝐼𝐼𝐼 und 𝐼𝐼𝐼𝐼 𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 26 26 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 unigung der Winkelgeschwindigkeit 𝜔𝜔𝜔𝜔 2 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

Description of the load cycle

3.3

Masses and moments of inertia

Electrical motors move mechanical elements. Their masses and moments of inertia determine the motor torque required for the movement. 3.3

Massen und TrĂ¤gheitsmomente

The masses and moments of inertia of Elemente. the mechanical elements determined by the material Elektrische Antriebe bewegen mechanische Deren Massen undare TrĂ¤gheitsmomente

bestimmen dastheir fĂźr die Bewegung erforderliche Motordrehmoment. used and by geometrical form. In case of rotational movements, the position of the rotatioDie Massen und TrĂ¤gheitsmomente der mechanischen Elemente werden vom verwendeten Material

nal axis additionally hasForm an influence moment of inertia. sowie ihrer geometrischen bestimmt.on Beithe Rotationsbewegungen hat zusĂ¤tzlich die Lage der Rotationsachse einen Einfluss auf das TrĂ¤gheitsmoment.

The masses and moments of inertia of the most important geometric basic bodies are given Nachfolgend sind die Massen und der wichtigsten geometrischen below. In the determination of TrĂ¤gheitsmomente the moment of inertia, it is assumed that theGrundkĂśrper rotational axis runs angegeben. Bei der Bestimmung des TrĂ¤gheitsmoments wird davon ausgegangen, dass die

through the gravitational centre of the body. verlĂ¤uft. Rotationsachse durch den Schwerpunkt desbasic GrundkĂśrpers KĂśrper Body

Graphical Darstellung

representation

Masse Mass m

TrĂ¤gheitsmoment Moment of inertia J

PunktPoint masse mass

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;

Cylinder Zylinder

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

Zylinder Cylinder

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

HohlHollowzylinder cylinder

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

Kegel Cone

cone

Kugel Sphere

Quader

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;(đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;1 + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;2 )đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 2 â&#x2C6;&#x2019; đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 2 ďż˝ 12

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;(đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;1 + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;2 )đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 4 â&#x2C6;&#x2019; đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 4 ďż˝ 160

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; 4 â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; 4 ďż˝ 32

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

1 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2 4 đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; 4 + đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; 3 â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2 64 48

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; 2 â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; 2 ďż˝ 4

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

Trun-

Kegelcated stumpf

đ??˝đ??˝đ??˝đ??˝ =

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

1 đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?[đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? 2 ] 12

Kugel

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

Cuboid Quader

J S:

Descriptionđ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; of the load cycle

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

TrĂ¤gheitsmoment, bei dem Moment of inertia, in which the durch axisdie of Rotationsachse rotation passes through theden centre of gravity verlĂ¤uft Schwerpunkt

m: Ď : l

Masse Mass Dichte Density Length LĂ¤nge

đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; =

đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; 5

Durchmesser d: Diameter Breite a: Width HĂśhe b: Height

27 Table 19 Mass and moment of inertia of geometric basic bodies

Real mechanical elements are often realised by a combination of geometrical basic bodies. In this case, the mechanical element is decomposed into its basic bodies and the mass and the moment of inertia of the mechanical element are calculated in 3 steps: Tabelle 19 Masse und TrĂ¤gheitsmoment geometrischer GrundkĂśrper 1. Determination the masses of inertia all basic bodies Reale mechanische of Elemente sind oftand ausmoments einer Kombination vonofgeometrischen GrundkĂśrpern aufgebaut. In diesem Fall wird das mechanische Element in seine GrundkĂśrper zerlegt und die Masse It should be noted that the rotational axis, which is set during the determination of the moments und das TrĂ¤gheitsmoment des mechanischen Elements in 3 Schritten berechnet: of inertia of the basic body, is parallel to the actual rotational axis of the mechanical element. Ermittlung der Massen und TrĂ¤gheitsmomente aller GrundkĂśrper 2. A1. pplication of Steiner's theorem Dabei ist zu beachten, dass die Rotationsachse, die bei der Ermittlung der TrĂ¤gheitsmomente Application of Steiner's theorem Forparallel all thezur basic bodies whose rotational des axismechanischen in the first der GrundkĂśrper angesetzt wird, tatsĂ¤chlichen Rotationsachse Elements liegt. calculation step has a parallel offset to the rotational axis of the mechanical element, the 2. Anwendung des Satzes von Steiner moment of inertia is determined with reference to im the1.actual rotational axis of the mechanical FĂźr alle die GrundkĂśrper, deren Rotationsachse Berechnungsschritt einen parallelen Versatz zur Rotationsachse des mechanischen Elementes aufweist, wird das element. TrĂ¤gheitsmoment bezogen auf die tatsĂ¤chliche Rotationsachse des mechanischen Elements 3. Addition of all masses and moments of inertia ermittelt. 3. Addition allermoments Massen und TrĂ¤gheitsmomente The masses and of inertia of the basic bodies determined in the first and second Die im 1. und 2. Schritt ermittelten Massen und TrĂ¤gheitsmomente der GrundkĂśrper werden stepsaddiert are added and give mass and the moment of inertia the mechanical element. und ergeben diethe Masse und das TrĂ¤gheitsmoment des of mechanischen Elements. Satz von theorem Steiner Steiner's

TrĂ¤gheitsmoment Moment of inertia J

đ??˝đ??˝đ??˝đ??˝ = đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 2 J: m:

TrĂ¤gheitsmoment Moment of inertia Masse Mass

Tabelle 20 Satz theorem von Steiner Table 20 Steiner's

d: J S:

Abstandofder parallelen Distance rotational axes Rotationsachsen Moment of inertia, in which the rotational axis passes TrĂ¤gheitsmoment, bei dem die Rotationsachse through the centre of gravity verlĂ¤uft durch den Schwerpunkt

Zurorder Bestimmung der Massen und TrĂ¤gheitsmomente die Dichte des Materials, aus dem diegeoIn to determine the masses and moments ofmuss inertia, the density of the material the geometrischen GrundkĂśrper bestehen, bekannt sein. FĂźr einige hĂ¤ufig verwendete metric basic bodies consist of must be known. For some frequently used construction materiKonstruktionswerkstoffe sind nachfolgend die Werte fĂźr ihre Dichte angegeben. als, the values for their density are given below. Material Acryl Aluminium Bronze

Dichte Ď 1190 kg/mÂł 2700 kg/mÂł 8100 kg/mÂł

Description of the load cycle Material

Density Ď

Acrylic

1190 kg/mÂł

Aluminium

2700 kg/mÂł

Bronze

8100 kg/mÂł

Oak wood

670 kg/mÂł

Spruce

470 kg/m3

Glass

2500 kg/mÂł

Cast iron

7600 kg/mÂł

Rubber

950 kg/mÂł

Laminated paper

1400 kg/mÂł

Copper

8940 kg/mÂł

Brass

8500 kg/mÂł

Polyethylene

930 kg/mÂł

Polyvinylchloride

1300 kg/mÂł

Steel

7900 kg/mÂł

Table 21 Density of frequently used construction materials

3.4

Load forces and load torques

In the mechanical system work is performed. This leads to load forces, which have be applied by the electrical motor as motor torque. If the load force acts on OberflĂ¤che the surfacerotierender of rotating mechanicalElementen elements,an, thelĂ¤sst effective load das torque Greift die Lastkraft an der mechanischer sich damit wirksame Lastdrehmoment entsprechend 5 ermitteln. can be determined according to FigureAbbildung 5.

Name Name

Darstellung Graphical representation

Lastkraft Load force

Zugkraft Traction force

đ??šđ??šđ??šđ??šđ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?

Weight force,

Gewichtskraft, downhill slope Hangabtriebskraft

force

Gleitreibung

đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??ş = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161; sin(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) = đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;

đ??şđ??şđ??şđ??ş

â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; cos(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ )

FĂźr vâ&#x2030; 0 wirkt FRg entgegen der

Gewichtskraft, Hangabtriebskraft

Description of the load cycle

= đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;

Sliding friction Gleitreibung

đ??şđ??şđ??şđ??ş

â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; cos(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ )

FĂźr entgegen der Forvâ&#x2030; vâ&#x2030; 0 wirkt 0, FRgFRg acts against the aktuellen Geschwindigkeit.

actual velocity. đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x2026; =

Haftreibung Static friction

đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;

đ??ťđ??ťđ??ťđ??ť

â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; cos(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ )

= đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; cos(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ )

entgegen der FĂźr wirkt Forv= v=0 0, FRhFRh acts against the aktuellen KraftF.F. actual force

Rolling friction Rollreibung

der FĂźr Rr entgegen Forvâ&#x2030; vâ&#x2030; 0 wirkt 0, FRrFacts against the aktuellen Geschwindigkeit. actual velocity.

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; = đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; â&#x2C6;&#x2122; đ??´đ??´đ??´đ??´

StrĂśmungsFlow resistance widerstand

đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; â&#x2C6;&#x2122; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł 2 2

Forvvâ&#x2030; FSt Facts against the FĂźr â&#x2030; 00, wirkt St entgegen der actual velocity. aktuellen Geschwindigkeit.

StrĂśmungsFlow resistance widerstand fĂźr for alaminare laminar flow eine in a tube in einem StrĂśmung Rohr Load force Lastkraft F: Mass to be moved Zu bewegende Masse m: Erdbeschleunigung Acceleration due to gravity g: Neigungswinkel Angle of inclination Îł: đ?&#x203A;˝: Gleitreibungskoeffizient Sliding friction coefficient ÂľG : Haftreibungskoeffizient Static friction coefficient ÂľH: Rollwiderstandskoeffizient cR : Rolling friction coefficient Tabelle 22 Typische LastkrĂ¤fte

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; = 8đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; â&#x2C6;&#x2122; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł

ForFĂźr vâ&#x2030; v0, actsFStagainst the â&#x2030; 0FSt wirkt entgegen der Geschwindigkeit. actualaktuellen velocity. l: v: Îˇ: Ď : cW: A:

Length theRohrs tube LĂ¤nge of des Geschwindigkeit Velocity ViskositĂ¤t zu fĂśrdernden Viscosity of des the flowing mediumMediums Dichte of des fĂśrdernden Mediums Density thezumedium StrĂśmungswiderstandskoeffizient Drag coefficient BezugsflĂ¤che (z.B. face StirnflĂ¤che eines Reference area (e.g. of a vehicle, Fahrzeugs, inner surface InnenflĂ¤che of a tube) eines Rohrs)

Table 22 Typical load forces

The following table shows some typical values of friction coefficients. These should be checked and adjusted according to the real application.

Sliding friction coefficients ÂľG

Static friction coefficients ÂľH

Rolling friction coefficients cR

Steel on steel without lubrication

0,12

0,15

0,0005 - 0,0010 (Ball bearing)

Steel on steel with lubrication

0,01

0,001 - 0,002 (Railway wheel on rail)

Nachfolgend sind Richtwerte fĂźr einige Reibungskoeffizienten angegeben. Diese sollten fĂźr konkrete Anwendungen geprĂźft undNachfolgend ggf. angepasst werden.fĂźr sind angegeben. Diese Diese sollten sollten fĂźr fĂźr konkrete konkrete Nachfolgend sind Richtwerte Richtwerte fĂźr einige einige Reibungskoeffizienten Reibungskoeffizienten angegeben. Description of the load cycle Anwendungen geprĂźft und ggf. angepasst werden. Anwendungen geprĂźft und ggf. angepasst werden.

Gleitreibungskoeffizienten ÂľG Haftreibungskoeffizienten ÂľH Rollwiderstandskoeffizienten cR ÂľHH Rollwiderstandskoeffizienten Rollwiderstandskoeffizienten ccRR Gleitreibungskoeffizienten Stahl auf Stahl Stahl auf StahlÂľÂľGGG Haftreibungskoeffizienten Kugellager Haftreibungskoeffizienten Âľ Gleitreibungskoeffizienten H R 0,12 0,15 0,0005 - 0,0010 Steel on Bronze 0,01 - 0,02 (Tires on ohne Schmierung ohne Schmierung (Stahl auf Stahl) 0,18 0,19 Stahl auf Stahl Stahl auf Stahl Kugellager Stahl auf Stahl Stahl auf Stahl Kugellager without lubrication asphalt or concrete) 0,12 0,15 0,0005 - 0,0010 0,12 0,15 Stahl auf Stahl Stahl auf Stahl Eisenbahnrad ohne ohne (Stahl auf auf Stahl) Stahl) 0,0005 - 0,0010 ohne Schmierung Schmierung ohne Schmierung Schmierung (Stahl 0,01 0,01 0,001 - 0,002 mit Schmierung mit Schmierung auf Schiene Stahl auf Stahl Stahl auf Stahl Eisenbahnrad Stahl auf Stahl Stahl auf Stahl Eisenbahnrad 0,01 0,01 0,001 -- 0,002 0,002 Steel on Bronze 0,01 0,01 0,001 mit mit auf Schiene Schiene 0,07 mit Schmierung Schmierung mit Schmierung Schmierung Reifen auf auf withauf lubrication Stahl Bronze Stahl auf Bronze Reifen auf 0,18 0,19 Stahl Stahl Bronze Asphalt oderReifen auf 0,01 - 0,02 Stahl auf auf Bronze Bronze Stahl auf auf Bronze ohne Schmierung ohne Schmierung Asphalt oder oder 0,01 -- 0,02 0,02 0,18 0,19 Asphalt 0,01 0,18 0,19 ohne ohne ohne Schmierung Schmierung ohne Schmierung Schmierung Beton Beton Beton Table 23 Typical values of friction coefficients Stahl auf Bronze Stahl Stahl auf auf Bronze Bronze 0,07 0,07 0,07 mit mit Schmierung mit Schmierung Schmierung Tabelle 23 Typische Reibungskoeffizienten Tabelle 23 Typische Reibungskoeffizienten Tabelle 23 Typische Reibungskoeffizienten The load force that emanates from flowing media is determined by its viscosity. In real applications,

this must be determinedDie asLastkraft, a function ofstrĂśmenden the temperature. die Medien ausgeht, wird wird wesentlich wesentlich von von ihrer ihrer ViskositĂ¤t ViskositĂ¤t bestimmt. bestimmt. In In Die Lastkraft, die von von strĂśmenden Medien ausgeht, der Applikation muss diese in von der Temperatur Temperatur ermittelt werden. Die Lastkraft, die von strĂśmenden Medien ausgeht, von ihrer ViskositĂ¤termittelt bestimmt. In der konkreten konkreten Applikation muss wird diese wesentlich in AbhĂ¤ngigkeit AbhĂ¤ngigkeit von der werden. der konkreten Applikation muss diese in AbhĂ¤ngigkeit von der Temperatur ermittelt werden. Material Temperature Ď&#x2026; Viscosity Îˇ Material Temperatur Material Temperatur Ď&#x2026;Ď&#x2026; Material Temperatur Ď&#x2026; ViskositĂ¤t Îˇ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Wasser 20 1 = 0,0001 0,0001 Wasser Water 20Â°C â&#x2C6;&#x2122;â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 20 Â°C Â°C 1 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ Wasser 20 Â°C 1 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = 0,0001 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; MotorĂśl 150 = 0,0003 0,0003 MotorĂśl 3 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122;â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Motor oil 150Â°C 150 Â°C Â°C đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; MotorĂśl 150 Â°C 3 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = 0,0003 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛â&#x2C6;&#x2122;â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = MotorĂśl 25 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = 0,01 0,01 MotorĂśl 100 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 25 Â°C Â°C Motor oil 25Â°C MotorĂśl

25 Â°C

100 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = 0,01

Table 24 Typical values of viscosity

Tabelle 24 Typische ViskositĂ¤ten

3.5

Example: Pump motor

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đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛

3.5 Beispiel: Pumpenantrieb The pump motor of the heating system runs 18 hours each day and is switched off during the Der Pumpenantrieb der Heizungsanlage lĂ¤uft wĂ¤hrende der Heizperiode jeweils 18 Stunden und wird night forder the purpose of energy for 6 hours.fĂźr The speed of the pump should be approx. wĂ¤hrend Nacht zum Zweck der saving Energieeinsparung 6 Stunden abgeschaltet. Die Drehzahl der Pumpe soll ca. 3000 1/min betragen. 3000 rpm. This results in the following motion sequence for the pump motor. Damit ergibt sich fĂźr den Pumpenantrieb der folgende Bewegungsablauf.

nMM::

Motordrehzahl Motordrehzahl

nM: Motor speed n M: Motordrehzahl

Tabelle 25 25 Bewegungsablauf Bewegungsablauf fĂźr Tabelle fĂźr einen einen Pumpenantrieb

On/Off

3.6 Beispiel: Beispiel: TĂźrantrieb TĂźrantrieb 3.6 FĂźr den den Bewegungsablauf Bewegungsablauf des des TĂźrantriebs TĂźrantriebs sind FĂźr sind folgende folgende Parameter Parameter gegeben: gegeben:

4 am

10 pm 12pm

Tabelle 25 Bewegungsablauf fĂźr einen Pumpenantrieb

30 30

Table 25 Motion sequence of a pump motor

3.6 Beispiel: TĂźrantrieb 3.6 Example: Door motor FĂźr den Bewegungsablauf des TĂźrantriebs sind folgende Parameter gegeben: 1,0 m â&#x20AC;˘ Verfahrweg (Breite der TĂźr) sPos: For the motion sequence of the door motor the following parameters are given: 0,5 m/sÂ˛ â&#x20AC;˘ Maximale Beschleunigung amax: Travel distance (width of the door) đ?&#x2018;&#x17D;max: 0.5 m / sÂ˛, Maxi0,4 m/s â&#x20AC;˘ Maximale Geschwindigkeit vmaxS: Pos:: 1.0 m, Maximum acceleration : 5 s â&#x20AC;˘ Minimale Pause zwischen zwei PositioniervorgĂ¤ngen T IV mum velocity Vmax: 0.4 m / s, Minimum pause between two positioning sequences TIV 5 s

Maximale Geschwindigkeit vmax: Minimale Pause zwischen zwei PositioniervorgĂ¤ngen TIV:

0,4 m/s 5s Description of the load cycle Die BewegungsvorgĂ¤nge zum Ă&#x2013;ffnen und SchlieĂ&#x;en der TĂźr verlaufen geschwindigkeitsoptimal mit den gleichen Bewegungsparametern. Deshalb ist es ausreichend, lediglich den Vorgang zum Ă&#x2013;ffnen der TĂźr zu betrachten. The movements for opening and closing the door are time-optimised using the same motion parameVerwendet werden die mathematischen Gleichungen nach Tabelle 15. Damit ergibt sich fĂźr den ters. Therefore, it is sufficient to treat the operation for opening the door only. The mathematical equaTĂźrantrieb der folgende Bewegungsablauf. tions according to Table 15 are used. This results in the following motion sequence of the door motor. â&#x20AC;˘ â&#x20AC;˘

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đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł 1 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 0,4 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛ đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = + = + = 3,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122; 0,5 đ?&#x2018; đ?&#x2018; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 10,4 đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018; đ?&#x2018; 0.4 đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018; đ?&#x2018; Â˛ đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = + = + = 3.3 đ?&#x2018; đ?&#x2018; đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 0.4 đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122; 0.5 đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź = 5,0 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

the acceleration T I, TIII: Duration Dauer derofBeschleunigungsund deceleration derofBremsphase and phasesand T I, TIII: Duration the acceleration deceleration phases Duration of the constant Dauer der Konstantfahrtphase T II: speed phase T II: Duration of the constant speed phase Positionierdauer, des TPos: Positioning time, Zeitdauer duration of the gesamten Positioniervorgangs entire positioning sequence TPos: Positioning time, duration of the entire positioning sequence Dauer derofPause T IV: Duration pause

T IV: Duration of pause Verfahrweg Travel distance s: a: Geschwindigkeit v: Velocity s: Travel distance a: Tabelle Velocity 26 Bewegungsablauf fĂźr einen TĂźrantrieb v:

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź = 5.0 đ?&#x2018; đ?&#x2018; Beschleunigung Acceleration Acceleration

Table 26 Motion sequence of a door motor

Table 26 Motion sequence of a door drive 4 Bewegungswandlung mit Getrieben 4Getriebe Motion with gears Sie wandeln die motorseitigen BewegungsgrĂśĂ&#x;en Drehzahl und sindconversion mechanische Wandler. 4 Motion conversion withconvert gears Gearboxes aresomechanical converters. They speed and the torque of the motor in such a Drehmoment um, dass sie den Anforderungen derthe Applikation entsprechen. Dieser Prozess way that they meet the requirements of the application. This process includes the conversion of the schlieĂ&#x;t die Wandlung der Bewegungsart von einer rotatorischen Motorbewegung in eine motion type from a rotational movement into a translatory movement of the load. translatorische Bewegung dermotor Last mit ein. Gearboxes are mechanical converters. Theytrain convert the speed the torque of the motor If the load performs a linearBewegung movement, thewird drive including theinklusive loadand is referred as Linearachse a linear axis. FĂźhrt die Last eine lineare aus, der Antriebsstrang der Lasttoals If the movement of Bewegung the loadmeet is der rotational, the drive train the load is referred to as a rotatory in such a way that they the requirements of including the Antriebsstrang application. This process includes bezeichnet. Ist die Last rotatorisch, wird der inklusive der Last als axis. Rotationsachse bezeichnet. the conversion of the motion type converters from a rotational motor movement into a translatory Often, application-specific mechanical are included in the machines. They are treated mathematically as gears. movement of the load. 31

Each gearbox has a driving side and a driven side. There are mathematical relationships between the physical values of the driving side and the driven side. In case of constant transmission gears, these relationships are linear. In case of variable transmission gears they are not linear.

Motion conversion with gears If the load performs a linear movement, the motor train including the load is referred to as a linear axis. If the movement of the load is rotational, the motor train including the load is referred to as a rotating axis. Often, application-specific mechanical converters are included in the machines. They are treated mathematically as gears. Each gearbox has a driving side and a driven side. There are mathematical relationships between the physical values of the driving and driven side. In case of constant transmission gears, these relationships are linear. In case of variable transmission gears they are not linear. In many applications, several gears are arranged one behind the other. The first gear often converts the motion type from translational to rotational and a downstream rotational gearbox adjusts the speed, the torque as well as the geometrical direction of the axes. Mechanical friction occurs in gears. Part of the energy fed into the gearbox is converted into heat and thus lost. If the energy flows from the motor to the working machine, the motor must cover the losses in the gear. If the energy flows from the working machine to the motor (e.g. during braking), the working machine covers the losses in the gear. Therefore different equations for the calculation Motor operation

Generator operation

dary Boundary conditions:conditions: F > 0, v > 0F > 0, v > 0 Boundary conditions: or or F > 0, v > F 0< or0,F v< < 0, 0vF<<0t0, v < 0 or ď ˇM >M 0< 0, or > 0, or M > 0, M Ď&#x2030;> > 00,or Ď&#x2030; <ď ˇ0> 0 or M < 0, ď ˇ < M 0 < 0, ď ˇ < 0 or

Boundary Boundary conditions:conditions: F > 0, v < 0F > 0, v < 0 Boundary conditions: or F > 0, v < 0or or F < 0, vF><00, v > 0F < 0, v > 0 or > 0, 0 > 0, ď ˇ < 0 or M > 0, Ď&#x2030;or< 0 or M M < 0, Ď&#x2030; >ď ˇ0< M or M < 0, ď ˇ > M 0 < 0, ď ˇ > 0 or M: Torque

F: Force

Force M: Torque F: Force M: Torque v: Velocity Ď&#x2030;: Angular speed Velocity v: Velocity ď ˇ: Angular ď ˇ: speed Angular speed Table 27 Energy flow atat motor and and generator operation operation 27 Table Energy at motor and generator operation 27flow Energy flow motor generator

4.1 Rotating axes 4.1.1 Rotating load Rotatory axes 4.1 Rotatory axes Rotatory load 4.1.1 Rotatory load toryAload occurs if forces are applied theapplied surface a surface rotating or a roller rotating work piece. rotatory load occurs if forces aretoare applied to the surface ofofaroller rotating work piece. A rotating load occurs if forces to of the arotating rotating rolleror or aa work y tables are also among the rotatory loads. Rotary tables are also among the rotatory loads. piece. Rotating tables are also among the rotating loads.

ory Rotatory load load 38

Equations Equations n:

Speedn:driving side Speed driving side

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; =

đ?&#x153;&#x201D;đ?&#x153;&#x201D; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; =

đ?&#x153;&#x201D;đ?&#x153;&#x201D; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;

4.1 Rotationsachsen 4.1.1 Rotatorische Last Eine rotatorische Last tritt immer dann auf, wenn an der OberflĂ¤che einer rotierenden Walze oder conversion with gears eines rotierenden WerkstĂźcks KrĂ¤fte angreifen. Drehtische zĂ¤hlen ebenfalls zuMotion den rotatorischen Lasten. Rotatorische Last Rotating load

Gleichungen Equations n:

Drehzahl Antriebsseite Speed driving side (1/s)

driving side M: Torque Drehmoment Antriebsseite P:

Leistung Antriebsseite Power driving side

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; =

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝

Winkel Antriebsseite Abtriebsseite đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;: Angle driving side and und driven side Winkelgeschwindigkeit Antriebsseite und Abtriebsseite đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;: Angular speed driving side and driven side Winkelbeschleunigung Antriebsseite und Abtriebsseite acceleration driving side and driven side đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź: Angular Moment of inertia load TrĂ¤gheitsmoment der Last J: Diameter load der Last Durchmesser d: Tabelle 28 Gleichungen der rotatorischen Last

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + đ??šđ??šđ??šđ??šđ??żđ??żđ??żđ??ż đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;

Table 28 Equations of the rotating load 32

4.1.2 Pump A pump is defined by an experimentally determined pump characteristic Î&#x201D;pP(Q). This characteristic is valid at a fixed speed of the pump as well as at a defined temperature and a defined conveying medium. Plants which are supplied with the volume flow produced by the pump have a system characte-

4.1.2 Pumpe Î&#x201D;pA(Q). In general, the system characteristic has the shape of a parabola, which has been 4.1.2 ristic Pump Eine Pumpe ist durch eine experimentell bestimmte Pumpenkennlinie â&#x2C6;&#x2020;pP(Q) definiert. Diese Kennlinie A pump is defined by an from experimentally determined pump characteristic ď &#x201E;pP(Q). Thispump characteristic is where shifted the coordinate Theeiner operating point of the is located wird bei upwards einer festen Drehzahl der Pumpeorigin. sowie bei definierten Temperatur und einem valid at a fixed speed of the pump as well as at a defined temperature and a defined conveying definierten FĂśrdermedium ermittelt. the pump characteristic intersects the system characteristic. At this operating point, the volume medium. Anlagen, die mit dem von der Pumpe produzierten Volumenstrom versorgt werden, weisen eine Plantsflow which supplied with the volume flow produced by the pump have a system characteristic is are determined. â&#x2C6;&#x2020;pA(Q) auf. Im Allgemeinen verlĂ¤uft die Anlagenkennlinie als Parabel, die vom Anlagenkennlinie ď &#x201E;pA(Q).Koordinatenursprung In general, the system characteristic has thewurde. shape of a parabola, which has been shifted nach oben verschoben upwards from the coordinate origin. Deraddition Arbeitspunkt derpump Pumpe stellt sich dort ein, wo die Pumpenkennlinie die Anlagenkennlinie In to the characteristic, pumps also have an efficiency characteristic Îˇ(Q). If the The operating point the pump is located where the pump characteristic schneidet. An of diesem Arbeitspunkt wird die FĂśrderleistung ermittelt. intersects the system volume At flow known at the operating required input power at the shaft of the pump characteristic. thisisoperating point, the volume point, flow is the determined.

can beder determined. Neben Pumpenkennlinie verfĂźgen Pumpen zusĂ¤tzlich Ăźber eine Wirkungsgradkennlinie Îˇ(Q). Ist In addition to the pump characteristic, pumps also kann have damit an efficiency characteristic ď ¨(Q). If the volume die FĂśrderleistung am Arbeitspunkt bekannt, die erforderliche Eingangsleistung an der flow isWelle knownder at Pumpe the operating point, the required input power at the shaft of the pump can be ermittelt werden. determined. Pump

ď &#x201E;pA: ď &#x201E; p P:

Pumpe Pump

Gleichungen Equations Equations

der Anlage â&#x2C6;&#x2020;pA: Druckdifferenz Pressure difference of the plant (Anlagenkennlinie) fĂźr n=konstant (system characteristic) for n = constant â&#x2C6;&#x2020;pP: Druckdifferenz der Pumpe Pressure(Pumpenkennlinie) difference of the pump fĂźr n=konstant (pump characteristic) for n = constant

đ??´đ??´đ??´đ??´ = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;) Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ??´đ??´ = Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? = đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?1 = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;) Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?2đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?1 2= đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

Motion conversion with gears Druckdifferenz derofAnlage difference the plant â&#x2C6;&#x2020;pA: Pressure

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´ = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

(Anlagenkennlinie) fĂźrfor n=konstant (system characteristic) n = constant

Druckdifferenz derofPumpe difference the pump â&#x2C6;&#x2020;pP: Pressure

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?1 = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

(Pumpenkennlinie) (pump characteristic)fĂźr for n=konstant n = constant

FĂśrderhĂśhe derofPumpe Discharge head the pump (Pumpenkennlinie) (pump characteristic)fĂźr for n=konstant n = constant

Îˇ:

Wirkungsgradkennlinie Efficiency characteristic ofder the pump Pumpe fĂźr n=konstant for n = constant

Drehzahl Antriebsseite Speed driving side (1/s)

PP:

FĂśrderleistung derof Pumpe Conveying capacity the pump

Leistung Antriebsseite Power driving side

Drehmoment Antriebsseite Torque driving side

Druck am Pumpe p1: Pressure at Eingang the inlet der of the pump Druck am Pumpe p2: Pressure at Ausgang the outletder of the pump Druckdifferenz am at Arbeitspunkt difference operating point â&#x2C6;&#x2020;pAP: Pressure Density medium to be pumped Dichte of des zu fĂśrdernden Mediums Ď : Tabelle 29 Gleichungen der Pumpe Table 29 Equations of the pump

đ??ťđ??ťđ??ťđ??ť =

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;) Ď â&#x2C6;&#x2122;g

đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

constant n=konstant

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; â&#x2C6;&#x2122; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; =

g: Q: QAP:

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; â&#x2C6;&#x2122; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ??´đ??´đ??´đ??´đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Erdbeschleunigung Acceleration due to gravity Volumenstrom zu fĂśrdernden Mediums Volume flow of des the medium to be pumped Volumenstrom zu fĂśrdernden Mediums Volume flow of des the medium to be pumped at am Arbeitspunkt operating point đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; =

4.1.3 Fan 4.1.3 LĂźfter

Ein LĂźfter ist durch eine experimentell bestimmte LĂźfterkennlinie â&#x2C6;&#x2020;pL(Q) definiert. Diese Kennlinie wird bei einer festen Drehzahl des LĂźfters sowie bei einer definierten Temperatur und Luftdichte ermittelt.

A fan is defined by vom an experimentally determined fan versorgt characteristic (Q). eine This characteristic pL Anlagen, die mit dem LĂźfter produzierten Volumenstrom werden, Î&#x201D; weisen auf.speed Im Allgemeinen verlĂ¤uft dieas Anlagenkennlinie als Parabel durch Anlagenkennlinie is determined atâ&#x2C6;&#x2020;paA(Q) fixed of the fan as well at a defined temperature andden air density. Koordinatenursprung.

Plants which are supplied with the volume flow produced by the fan have a system characte33

ristic Î&#x201D;pA(Q). In general, the system characteristic has the shape of a parabola starting at the coordinate origin. The operating point of the fan is located where the fan characteristic intersects the system characDer Arbeitspunkt des LĂźfters stellt sich dort ein, wo die LĂźfterkennlinie die Anlagenkennlinie schneidet.

teristic. At Arbeitspunkt this operating point, the volume is characteristic determined. intersects In addition the fan characteristic, AnThe diesem wird ermittelt. operating point of the fandie is FĂśrderleistung located where flow the fan theto system characteristic. At operatingcharacteristic point, the volume flow is determined. fans also anthis efficiency Îˇ(Q). IfĂźber the volume flow is known at the operating Îˇ(Q). Ist die point, Neben derhave LĂźfterkennlinie verfĂźgen LĂźfter zusĂ¤tzlich eine Wirkungsgradkennlinie

FĂśrderleistung am Arbeitspunkt bekannt, kann damit diebe erforderliche Eingangsleistung an der Welle the power at the shaft thehave fan an can determined. Inrequired addition toinput the fan characteristic, fansof also efficiency characteristic ď ¨(Q). If the volume flow des LĂźfters ermittelt werden.

is known at the operating point, the required input power at the shaft of the fan can be determined.

LĂźfter Fan Fan

â&#x2C6;&#x2020;pA: Druckdifferenz der Anlage ď &#x201E;pA: (Anlagenkennlinie) Pressure difference of the plant fĂźr n=konstant (system characteristic) for n = constant â&#x2C6;&#x2020;pL: Druckdifferenz des LĂźfters ď &#x201E;pP: (LĂźfterkennlinie) Pressure differencefĂźr of n=konstant the fan (fan characteristic) for n =des constant Wirkungsgradkennlinie LĂźfters Îˇ: ď ¨: Efficiency characteristic of the fan fĂźr n=konstant

Gleichungen Equations Equations

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ??´đ??´ =đ??´đ??´đ??´đ??´ đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ??żđ??ż = đ?&#x2018;?đ?&#x2018;?2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?1 = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;) đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

Motion conversion with gears Druckdifferenz derofAnlage â&#x2C6;&#x2020;pA: Pressure difference the plant

(system characteristic) n = constant (Anlagenkennlinie) fĂźrfor n=konstant Druckdifferenz des â&#x2C6;&#x2020;pL: Pressure difference of LĂźfters the fan (fan characteristic) for = constant (LĂźfterkennlinie) fĂźr nn=konstant Wirkungsgradkennlinie Îˇ: Efficiency characteristic ofdes the LĂźfters fan for = constant fĂźrn n=konstant

Drehzahl Antriebsseite Speed driving side (1/s)

Leistung Antriebsseite Power driving side

Drehmoment Antriebsseite Torque driving side

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??żđ??żđ??żđ??ż = đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?1 = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;) đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;)

constant n=konstant

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; =

Luftdruck Eingang p1: Air pressuream at the inlet ofdes the LĂźfters fan pressuream at the outlet of theLĂźfters fan Luftdruck Ausgang des p2: Air Tabelle 30 Gleichungen des LĂźfters Table 30 Equations of the fan

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ â&#x2C6;&#x2122; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; =

Volumenstrom der Luft Q: Volume Q: flow of the air QOP: Volume flow of the medium to be pumped at operating point

4.2 Linear axes 4.2.1 Spindle 4.2

Linearachsen

4.2.1 Spindel A spindle realises linear movement of the load in a limited travel range. The load is fixed so that Eine Spindel realisiert lineare Bewegungen der Last in einem begrenzten Verfahrbereich. Die Last ist

vertical is also possible. The required torque fixiert, somotion dass auch senkrechte Bewegungen mĂśglich sind.at the driving side can be reduced by means Durch einen Last- und Kraftausgleich kann das erforderliche Drehmoment an der Antriebsseite

of load and force compensation. reduziert werden. Spindel Spindle

Gleichungen Equations

Ď&#x2022;:

Winkel Antriebsseite Angle driving side

Ď&#x2030;:

Winkelgeschwindigkeit Angular speed driving side Antriebsseite

Drehzahl Antriebsseite Speed driving side (1/s)

Îą:

Winkelbeschleunigung Angular acceleration Antriebsseite driving side

Randbedingung: Boundary condition: 0â&#x2030;¤ đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ â&#x2030;¤2

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

Linear force Lineare Kraft

FH:

Hangabtriebskraft derthe Last, Downhill slope force of load, des Schlittens the slide and the und nut der Mutter

FB:

Beschleunigungskraft Acceleration force of the der load,Last, the slide and the und nut der Mutter des Schlittens

F R:

Friction force the guidance Reibkraft in in der FĂźhrung ofdes theSchlittens slide

Torque driving side, Drehmoment Antriebsmotor operation seite, motorischer Betrieb

Drehmoment AntriebsTorque driving side, seite, generatorischer generator operation Betrieb

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝ â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝ â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??š = đ??šđ??šđ??šđ??šđ??ťđ??ťđ??ťđ??ť + đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ + đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2019; đ??šđ??šđ??šđ??šđ??´đ??´đ??´đ??´ â&#x2C6;&#x2019; đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??şđ??´đ??´đ??´đ??´

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 1 ďż˝â&#x2C6;&#x2122; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; ďż˝ â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

v>0 v=0 v<0

2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 1 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122;

FA:

Kraft des Kraftausgleichs (Wird durch Applikation bestimmt.)

FGA:

Kraft des Gewichtsausgleichs (Ist nur einsetzbar, wenn dv/dt < g gilt.)

đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ??şđ??şđ??şđ??ş â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; (đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = â&#x2C6;&#x2019;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ??şđ??şđ??şđ??ş â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; (đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) đ??šđ??šđ??šđ??šđ??´đ??´đ??´đ??´ = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą)

đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??şđ??´đ??´đ??´đ??´ = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??şđ??şđ??şđ??şđ??´đ??´đ??´đ??´ [đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;]

v>0 F R:

Reibkraft in der FĂźhrung Schlittens Motion des conversion with gears

v=0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

v<0

Kraftofdes FA: Force the Kraftausgleichs force compensation (Is(Wird determined the application.) durch by Applikation bestimmt.)

đ??šđ??šđ??šđ??šđ??´đ??´đ??´đ??´ = đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą)

Kraftofdes FGA: Force the Gewichtsausgleichs weight compensation (Applicable, if dv/dt < g.) (Ist nur einsetzbar, wenn dv/dt < g gilt.) Weg Abtriebsseite s: Distance driven side Geschwindigkeit v: Velocity driven side Abtriebsseite Beschleunigung Abtriebsseite driven side a: Acceleration Steigung der Spindel of the spindle P: Pitch of the spindle Wirkungsgrad der Spindel Îˇ: Efficiency of inertia of the TrĂ¤gheitsmoment derspindle Spindel J: Moment Tabelle 31 Gleichungen der Spindel Table 31 Equations of the spindle

4.2.2

Neigung derspindle Spindelrelated gegenĂźber Horizontalen đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝: Incline of the to the der horizontal Gleitreibungskoeffizient friction coefficient ÂľG: Sliding of der the load, and theund nut der Mutter Masse Last,the desslide Schlittens mL: Mass of des the counterweight Masse Gegengewichts mGA: Mass Acceleration due to gravity Erdbeschleunigung g:

Conveyor belt

A conveyor belt realises linear movement of the load in a theoretically infinite travel range. 4.2.2 FĂśrderband Ein FĂśrderband realisiert lineare Bewegungen der Last in einem theoretisch unendlichen Verfahrbereich. Diefixed Last ist oft nur Ăźber die Haftreibung aufthe dem FĂśrderband fixiert, so dass The load is often only by static friction so that slope of the conveyor beltdie is limited. Steigung des FĂśrderbands begrenzt ist.

Conveyor belt FĂśrderband

Boundary Randbedingung: condition:

Equations Gleichungen 35

Ď&#x2022;:

Winkel Antriebsseite Angle driving side

Ď&#x2030;:

Winkelgeschwindigkeit Angular speed Antriebsseite driving side

Drehzahl Antriebsseite Speed driving side (1/s)

Îą:

Angular acceleration Winkelbeschleunigung driving side Antriebsseite

Drehmoment AntriebsM: Torque driving side, seite, motorischer motor operation Betrieb Drehmoment AntriebsM: Torque driving side, seite, generatorischer generator operation Betrieb

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

0â&#x2030;¤ đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ <2

LineareForce Kraft Linear

FH:

Hangabtriebskraft der Downhill slope force of Last the load

FB:

Acceleration force of the Beschleunigungskraft derload, Last, des the belt andder theStĂźtzrollen support rolls Gurts und

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = 2 â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = 2 â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??šđ??ťđ??ťđ??ťđ??ť = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = ďż˝đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??şđ??şđ??şđ??ş + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;

v>0 F R:

Reibkraft der of Last Friction force the load

s: v: a: dU: dS: nS: JU: J S:

Weg Abtriebsseite Geschwindigkeit Abtriebsseite Beschleunigung Abtriebsseite Durchmesser der Umlenkrollen Durchmesser der StĂźtzrollen Anzahl der StĂźtzrollen TrĂ¤gheitsmoment einer Umlenkrolle TrĂ¤gheitsmoment einer StĂźtzrolle

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; 2

ďż˝ â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

v=0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

v<0

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; 2 đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; 1 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; 2 đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122;

Îˇ: Î˛: ÂľG: mL: mG: g:

Wirkungsgrad des FĂśrderbandes Neigung des FĂśrderbandes gegenĂźber der Horizontalen Gleitreibungskoeffizient Masse der Last Masse des Gurts Erdbeschleunigung

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; 1 ďż˝â&#x2C6;&#x2122; 2 đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; ďż˝ â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 2

v>0 F R:

Reibkraft der Last

v=0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0 Motion conversion with gears đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = â&#x2C6;&#x2019;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ?&#x153;&#x2021;đ??şđ??şđ??şđ??ş â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; (đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝)

v<0 s: v: a: dU: dS: nS: JU: J S: Tabelle 32 Gleichungen des FĂśrderbands Weg Abtriebsseite Distance driven side Geschwindigkeit Velocity driven side Abtriebsseite Beschleunigung Acceleration driven side Abtriebsseite Diameter of the pulleys Durchmesser der Umlenkrollen Diameter of the support rolls Durchmesser der StĂźtzrollen Number theStĂźtzrollen support rolls Anzahlofder Moment of inertia of aeiner pulley TrĂ¤gheitsmoment Umlenkrolle Moment of inertia of aeiner support roll TrĂ¤gheitsmoment StĂźtzrolle

Îˇ: Î˛: ÂľG: mL: mG: g:

Wirkungsgrad FĂśrderbandes Efficiency of thedes conveyor belt Neigung FĂśrderbandes gegenĂźber Incline of des the conveyor belt related der Horizontalen to the horizontal Gleitreibungskoeffizient Sliding friction coefficient Masse Mass ofder theLast load Masse Mass ofdes the Gurts belt Erdbeschleunigung Acceleration due to gravity

Table 32 Equations of the conveyor belt

4.2.3

Wire rope and belt

4.2.3 Seilzug und Riemen Der Seilzug bzw. Riemen realisiert waagerechte Bewegungen einer Last in einem begrenzten The wire rope or belt realises horizontal movements of a load in a limited travel range. Verfahrbereich. Seilzug, Riemen Wire rope, belt

Gleichungen Equations

Ď&#x2022;:

Angle driving side Winkel Antriebsseite

Ď&#x2030;:

Winkelgeschwindigkeit Angular speed Antriebsseite driving side

Drehzahl Antriebsseite n:36 Speed driving side (1/s)

Îą: M: M:

F: FB:

Linear Force Lineare Kraft Acceleration force of the load, the Beschleunigungskraft der Last, rope or thebzw. belt des and Riemens the support des Seils undrolls der StĂźtzrollen

v>0 F R:

Reibkraft der of Last Friction force the load

v=0 v<0

Winkelbeschleunigung Angular acceleration driving side Antriebsseite Drehmoment AntriebsTorque driving side, seite, motorischer motor operation Betrieb Drehmoment AntriebsTorque driving side, seite, generatorischer generator operation Betrieb

đ??šđ??šđ??šđ??š = đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ + đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = ďż˝đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; 2

ďż˝ â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; ] â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;

Weg Abtriebsseite Wirkungsgrad Seilzugs s: Distance driven side Geschwindigkeit Efficiency of thedes wire rope or bzw. belt Riemens Îˇ: Abtriebsseite Beschleunigung driven side Rolling friction coefficient Rollwiderstandskoeffizient v: Velocity cR : driven side ofder the Last load Abtriebsseite Masse a: Acceleration mL: Mass of the der pulleys ofeiner a support roll Durchmesser Umlenkrollen Masse StĂźtzrolle dU: Diameter mS: Mass of the der support rolls ofdes the Seils rope or theRiemens belt Durchmesser StĂźtzrollen Masse bzw. dS: Diameter mSZ: Mass the support rolls Acceleration due to gravity Anzahl of der StĂźtzrollen Erdbeschleunigung nS: Number g: of inertia ofeiner a pulley TrĂ¤gheitsmoment Umlenkrolle JU: Moment of inertia of einer a support roll TrĂ¤gheitsmoment StĂźtzrolle JS: Moment Tabelle 33 Gleichungen des Seilzugs bzw. Riemens Table 33 Equations of the wire rope or belt

Motion conversion with gears

4.2.4

Toothed rack

4.2.4 Zahnstange A toothed rack realises horizontal movements of a load in a limited travel range. Eine Zahnstange realisiert waagerechte Bewegungen einer Last in einem begrenzten Verfahrbereich. Zahnstange Toothed Rack

Gleichungen Equations

Ď&#x2022;:

Angle driving side Winkel Antriebsseite

đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122;

Ď&#x2030; : Angular speed Winkelgeschwindigkeit driving side Antriebsseite n:

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122;

Drehzahl Antriebsseite

Speed driving side (1/s)

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122;

Winkelbeschleunigung acceleration Îą: Angular driving side Antriebsseite Drehmoment driving side, M: Torque motor operation Antriebsseite, motorischer Betrieb Drehmoment driving side, M: Torque generator operation Antriebsseite, generatorischer Betrieb F:

LineareForce Kraft Linear

FB:

Beschleunigungskraft derload, Last,the Acceleration force of the toothed rack andund the der support rolls der Zahnstange StĂźtzrollen

F R:

Reibkraft der of Last Friction force the load

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = ďż˝đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020; 2

ďż˝ â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

v=0

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

v<0 Weg Abtriebsseite s: Distance driven side Geschwindigkeit Velocity driven sideAbtriebsseite v: Beschleunigung Abtriebsseite Acceleration driven side a: Durchmesser Antriebsritzels of thedes driving pinion dU: Diameter Diameter of theder support rolls Durchmesser StĂźtzrollen dS: Number of the support rolls Anzahl der StĂźtzrollen nS: Tabelle 34 Gleichungen der Zahnstange

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;

1 đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??š = đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ + đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

v>0

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;

J S: Îˇ: cR : mL: mS: g:

TrĂ¤gheitsmoment StĂźtzrolle Moment of inertia ofeiner a support roll Wirkungsgrad Zahnstange Efficiency of theder toothed rack Rollwiderstandskoeffizient Rolling friction coefficient Mass the Last load and rack Masseofder und the dertoothed Zahnstange Mass a support roll Masseofeiner StĂźtzrolle Acceleration due to gravity Erdbeschleunigung

Table 34 Equations of the toothed rack

4.2.5 Hoist The hoist realises vertical movements of the load in a limited travel range. The torque needed at the driving side of the rope drum can be reduced by a combination of movable and fixed deflection pulleys.

4.2.5 Hubwerk Das Hubwerk realisiert senkrechte Bewegungen der Last in einem begrenzten Verfahrbereich. Ă&#x153;ber Motion conversion with gears die Kombination von losen und festen Umlenkrollen kann der Drehmomentbedarf an der Antriebsseite der Seiltrommel verringert werden.

Equations Gleichungen

Hubwerk Running gear

Anzahl der z: Number of supporting ropes tragenden Seile

đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??šđ??šđ??šđ??š + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??żđ??żđ??żđ??ż 2đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 2đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 2đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

Winkel Antriebsseite driving side Ď&#x2022;: Angle

đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122;

Winkelgeschwindigkeit Ď&#x2030;: Angular speed driving side Antriebsseite

Drehzahl n: Speed drivingAntriebsseite side (1/s) Winkelbeschleunigung Îą: Angular acceleration

driving side Antriebsseite Drehmoment AntriebsM: Torque driving side, motor operation seite, motorischer Betrieb Drehmoment M: Torque driving side,Antriebsgenerator operation seite, generatorischer Betrieb

Randbedingungen: Boundary condition: â&#x20AC;˘ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??šđ??šđ??šđ??š â&#x2030;Ľ 1; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??šđ??šđ??šđ??š â&#x2C6;&#x2019; 1 â&#x2030;¤ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??żđ??żđ??żđ??ż â&#x20AC;˘ The Masse des of Seils Hakens kĂśnnen mass theund ropedes and the hook can be neglected. vernachlĂ¤ssigt werden. F:

Lineare Kraft an rope der Seiltrommel Linear Force at the drum

Gewichtskraft der Last an rope drum FG: Weight force of the load at the der Seiltrommel Beschleunigungskraft der and Lastthe FB: Acceleration force of the load pulleys at the rope drum an der und der Umlenkrollen Seiltrommel Weg Abtriebsseite Geschwindigkeit s: Distance driven side Abtriebsseite driven sideBeschleunigung v: Velocity driven side Abtriebsseite a: Acceleration of the pulleys Durchmesser der Umlenkrollen dU: Diameter of the rope Durchmesser derdrum Seiltrommel dT: Diameter ofder movable pulleys Anzahl losendeflection Umlenkrollen nL: Number ofder fixed deflection pulleys, Anzahl festen Rollen, enthĂ¤lt die nF: Number contains rope drum rolls Seiltrommel undand die fixed festen Umlenkrollen Tabelle Gleichungen des Hubwerks Table 35 35 Equations of the hoist

4.2.6

đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??ş =

JU: J T: Îˇ: mL: mU: g:

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??š = đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??ş + đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 1 ďż˝â&#x2C6;&#x2122; 2 đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; ďż˝ â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 2

1 [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??żđ??żđ??żđ??ż đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; ] â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§

1 4đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§(đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ â&#x2C6;&#x2019; 1) đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = ďż˝ (đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??żđ??żđ??żđ??ż đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; ) + đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; ďż˝ â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; 2

TrĂ¤gheitsmoment Moment of inertia of aeiner pulleyUmlenkrolle TrĂ¤gheitsmoment derrope Seiltrommel Moment of inertia of the drum Efficiency of the rope Wirkungsgrad der drum Seiltrommel Mass of der the load Masse Last Mass of aeiner deflection pulley Masse Umlenkrolle Acceleration due to gravity Erdbeschleunigung

Running gear 4.2.6 Fahrwerk Ein Fahrwerk realisiert lineare Bewegungen der Last in einem theoretisch unendlichen Verfahrbereich.

A running gearbox realises linearerfolgt movements the load a theoretically infinite travel Die KraftĂźbertragung durch dieof Reibung derin angetriebenen RĂ¤der auf dem Untergrund. 4.2.6

Fahrwerk

Deshalb ist die Untergrunds begrenzt. range. The transmission of Steigung force is des done by the friction of the driven wheels on the ground. Ein Fahrwerk realisiert lineare Bewegungen der Last in einem theoretisch unendlichen Verfahrbereich. Die KraftĂźbertragung erfolgt durch die Reibung der angetriebenen RĂ¤der auf dem Untergrund. Therefore, the slope of the ground is limited.

Fahrwerk Deshalb ist die Steigung des Untergrunds begrenzt. Fahrwerk Running gear

Gleichungen Equations

Gleichungen

Ď&#x2022;:

Winkel Antriebsseite

Ď&#x2030;: Winkel2 Winkel Antriebsseite Ď&#x2022;: Angle driving side geschwindigkeit đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; Antriebsseite Ď&#x2030;: Angular Winkelspeed 2 Drehzahl driving side n: đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; geschwindigkeit đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; Antriebsseite Antriebsseite Boundary condition: 0â&#x2030;¤ đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ < Randbedingung: 2

Randbedingung:

0â&#x2030;¤ đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ < 2

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122;

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122;

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122;

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

1 đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

1 drivingÎąside Drehzahl : Winkeln:39 Speed 2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122; Antriebsseite beschleunigung đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; acceleration Antriebsseite Îą: Angular Winkel2 Drehmoment driving side M: beschleunigung đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝ â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ďż˝ Antriebsseite Antriebsseite, đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; 2 Drehmoment motorischer Betrieb M: đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; 1 Drehmoment M: Antriebsseite, đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; ďż˝ â&#x2C6;&#x2122; 2 đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; Antriebsseite, motorischer Betrieb đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; ďż˝ Drehmoment generatorischer M: 45 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; Antriebsseite, Betrieb

Îą:

Motion conversion with gears Boundary condition: 0â&#x2030;¤ đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝ < Randbedingung: 2

đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;

LineareForce Kraft Linear

FH:

Downhill slope force of Last, the load, the Hangabtriebskraft der running gear and des Chassis und the der wheels RĂ¤der

FB:

Beschleunigungskraft derload, Last,the Acceleration force of the des Chassis RĂ¤der running gear und and der the wheels

F R:

Reibkraft der of Last Friction force the load

2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ??˝đ??˝đ??˝đ??˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź + ďż˝đ??šđ??šđ??šđ??š â&#x2C6;&#x2122;

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; 1 ďż˝â&#x2C6;&#x2122; 2 đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ďż˝ â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 2

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ] â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; (đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = 0

Kraft des Gewichtsausgleichs Force of the weight compensation (Ist nur einsetzbar, (Applicable, if dv/dt wenn < g.) dv/dt < g gilt.) Weg Abtriebsseite Distance driven side Geschwindigkeit Velocity driven sideAbtriebsseite Beschleunigung Abtriebsseite Acceleration driven side Durchmesser RĂ¤der Diameter of theder wheels Anzahl der RĂ¤der Number of the wheels

4.3 4.3.1

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2122;

đ??šđ??šđ??šđ??šđ??ľđ??ľđ??ľđ??ľ = [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ] â&#x2C6;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

v=0

Tabelle 36 Gleichungen des Fahrwerks Table 36 Equations of the running gear

1 đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;

đ??šđ??šđ??šđ??šđ??ťđ??ťđ??ťđ??ť = [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ] â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝)

v<0

s: v: a: dR: nR:

Winkelbeschleunigung Antriebsseite Drehmoment Torque driving side, Antriebsseite, motor operation motorischer Betrieb Drehmoment Torque driving side, Antriebsseite, generator operation generatorischer Betrieb

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = đ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2122;

v>0

FGA:

Drehzahl Antriebsseite

đ??šđ??šđ??šđ??šđ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; [đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??żđ??żđ??żđ??ż + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ] â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; (đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝đ?&#x203A;˝) JR: cR : mL: mGA: mR: g:

đ??šđ??šđ??šđ??šđ??şđ??şđ??şđ??şđ??şđ??şđ??şđ??ş = đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ??şđ??şđ??şđ??şđ??şđ??şđ??şđ??ş â&#x2C6;&#x2122; [đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;]

TrĂ¤gheitsmoment eines Rades Moment of inertia of a wheel Rollwiderstandskoeffizient Rolling friction coefficient Masse derload Lastand undthe des Chassis Mass of the running gear Masse descounterweight Gegengewichts Mass of the Masse Mass of aeines wheelRades Acceleration due to gravity Erdbeschleunigung

Rotating gear Mathematical description and parameters

4.3 Rotationsgetriebe Rotating gears are used to adjust rotational motion variables and the shaft alignment bet4.3.1 Rotatory Mathematische Beschreibung und KenngrĂśĂ&#x;en 4.3 gear Rotationsgetriebe derworking Anpassung rotatorischer BewegungsgrĂśĂ&#x;en und der geometrischen 4.3.1 Mathematical description andmachine. parameters ween the motordienen and the Achsrichtungen zwischen Arbeitsmaschine. Rotatory gears are used to Motor adjustund rotational motion variables and the shaft alignment between the motor and the working machine. Rotationsgetriebe Rotating gear Rotatory gearbox

Gleichungen Equations Equations 40

ď Ş1:Ď&#x2022;1:

Winkel Antriebsseite Angle driving side Angle driving side

ď ˇ1:Ď&#x2030;1:

Winkelgeschwindigkeit Angular speed Angular speed driving side Antriebsseite driving side

n1: n1:

Drehzahl Speed drivingAntriebsseite side

ď Ą: Îą:

Winkelbeschleunigung Antriebsseite Angular acceleration driving side

M1 :

Drehmoment Antriebsseite,

đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; 1 1==đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; 2 2â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;

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đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2 11

Ď&#x2022;1:

Winkel Antriebsseite

Ď&#x2030;1:

Winkelgeschwindigkeit Antriebsseite

n1:

Drehzahl Antriebsseite Speed driving side

Îą:

Angular acceleration driving side Winkelbeschleunigung Antriebsseite

M1 :

Torque drivingAntriebsseite, side, Drehmoment motorischer Betrieb motor operation Torque drivingAntriebsseite, side, Drehmoment generatorischer Betrieb generator operation

M1 :

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Motion conversion with gears

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P1:

Leistung Antriebsseite Power driving side

n1N:

Nenndrehzahl der Antriebsseite Rated speed atan driving side

Die Drehzahl des Getriebes der Antriebsseite, The speed, the gearbox can auf provide at its driving side mit der es dauerhaft betrieben werden kann. permanently.

n1max:

Maximal zulĂ¤ssige Drehzahl Maximum permissible an derat Antriebsseite speed driving side

The maximum speed, the gearbox can provide at its mit der es maximal betrieben werden kann, driven side mechanische without mechanical destruction.auftreten. ohne dass BeschĂ¤digungen

M2N:

Nenndrehmoment Rated torque at driven side

The torque, the gearbox can provide permanently without ohne zusĂ¤tzliche KĂźhlmaĂ&#x;nahmen an seiner Abtriebsseite additional cooling measures at its driven side. Theals rated abgeben kann. Das Nenndrehmoment wird auch torque is also referred to as the permanent torque. Dauerdrehmoment bezeichnet.

Die Drehzahl des Getriebes auf der Antriebsseite,

Das Drehmoment, das das Getriebe bei einer dauerhaft

an der Abtriebsseite

permissible torque The torque, M2max: Maximum Das maximum Drehmoment, dasthe dasgearbox Getriebecan provide at its Maximal zulĂ¤ssiges Drehmoment maximal anwithout seiner Abtriebsseite abgeben kann. andriven der Abtriebsseite at side driven side mechanical destruction. Drehzahl Abtriebsseite Abtriebsseite Speed driven side Torque driven side n2: M2: Drehmoment Getriebeuntersetzung Angle side Gear ratio Winkeldriven Abtriebsseite i: GetriebeĂźbersetzung; Ď&#x2022;2: TrĂ¤gheitsmoment auf die Angular speed driven sideAbtriebsseite J: Moment of inertia ofdes theGetriebes gearbox related Winkelgeschwindigkeit Ď&#x2030;2: Antriebsseite Angular acceleration driven side to driving side bezogen Winkelbeschleunigung Abtriebsseite Îą2: Tabelle 37 Gleichungen des Rotationsgetriebes

Table 37 Equations of the rotating gearbox

Sind die BewegungsablĂ¤ufe an der Abtriebsseite des Getriebes bestimmt, wird ein konkretes Getriebe aus dem Katalog ausgewĂ¤hlt. Die Auswahl erfolgt unter BerĂźcksichtigung der mechanischen und To determineEigenschaften the motion sequences on die the mit output of the gearbox you have to select it from thermischen des Getriebes, Hilfe side von Kennlinien dargestellt werden.

the catalogue. The selection is made taking into account the mechanical and thermal properties of the which are by characteristic curves. 4.3.2gear, Auswahl derrepresented optimalen GetriebeĂźbersetzung

Die GetriebeĂźbersetzung ist so zu wĂ¤hlen, dass die maximal auftretende Abtriebsdrehzahl des Getriebes in eine Antriebsdrehzahl Ăźbersetzt wird, die von dem entsprechend Kapitel 1 gewĂ¤hlten 4.3.2 Selection Motortyp abgedeckt wird.of the optimum gear ratio 41 the maximum occurring speed at the driven The gear ratio has to be selected in a way that

side of the gearbox is translated into a speed which is covered by the motor type selected according to chapter 1. This results in a permissible range of gear ratio, which can be used for the optimization of the motor train. â&#x20AC;˘ For constant and variable speed motors, a small gear ratio should be selected. A small gear ratio often results in a high efficiency of the gearbox and reduces the losses of the motor train. â&#x20AC;˘ For servo motors, the gear ratio should be as close as possible to the optimum value iopt .

â&#x20AC;˘

GetriebeĂźbersetzung gewĂ¤hlt werden. Eine kleine GetriebeĂźbersetzung hat oft einen hohen Wirkungsgrad des Getriebes zur Folge und reduziert die Verluste des Antriebsstrangs. Motion conversion with gears Bei Servoantrieben sollte die GetriebeĂźbersetzung mĂśglichst nahe am optimalen Wert iopt liegen. â&#x20AC;˘ For constant and variable speed drives, a small gear ratio should be selected. A small gear ratio often results in a high efficiency the gearbox and of the drive train. gear ratio reduces the losses iopt: ofOptimum Optimaler Ă&#x153;bersetzungsfaktor des Getriebes đ??˝đ??˝đ??˝đ??˝2 â&#x20AC;˘ For servo be as as possible to the optimum value iopt. Summe TrĂ¤gheitsmomente auf derside Antriebsseite J1 : Sum of close theder moments of inertia at the driving đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153; drives, = ďż˝ the gear ratio should đ??˝đ??˝đ??˝đ??˝1 Summe TrĂ¤gheitsmomente auf der Abtriebsseite J2 Sum of theder moments of inertia at the driven side : Optimum gear ratio Tabelle 38 Optimale GetriebeĂźbersetzung fĂźrioptServoantriebe đ??˝đ??˝2 J1 : Sum of the moments of inertia at the driving side đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x153; = â&#x2C6;&#x161; đ??˝đ??˝ 1 Table 38 Optimum gear ratio for servo motors J2 of the moments of inertia at the driven side In diesem Fall ist das Drehmoment, das der Motor fĂźrSum die Realisierung des

Table 38 Optimum gear muss, ratio for drives Eine Wahl der GetriebeĂźbersetzung Bewegungsvorgangs aufbringen amservo niedrigsten. nahe am Optimum fĂźhrt im Auslegungsprozess zu einem kleineren Motor und einem is kleineren In this case, the motor torque needed for the realization of the motion sequence the lowest. StellgerĂ¤t. A selection ofcase, the gear ratio close the optimum leads to aofsmaller motor and a smaller control In this the motor torquetoneeded for the realization the motion sequence is the lowest. unit. A selection of the gear ratio close to the optimum leads to a smaller motor and a smaller control unit. 4.3.3 Auswahl des optimalen Getriebetyps 4.3.3 werden Selection of the optimum gearbox type Rotationsgetriebe in verschieden Typen angeboten. 4.3.3 Selection of the optimum gearbox type Rotatory are offered in various types. types. Rotatinggears gearboxes are offered in various

Rotating gearboxes

Abbildung 8 Getriebetypen Figure 8 Gearbox types Figure 8 Gearbox types

Jeder Getriebetyp weist spezifische StĂ¤rken und SchwĂ¤chen auf. Die Auswahl des optimalen Getriebetyps amtype einfachsten mit Hilfe eines Formulars entsprechend Tabellegearbox 39, die alle Eacherfolgt gearbox has its specific strengths and weaknesses. The optimum typeinis selected Each gearbox type has specific strengths and weaknesses. The optimum gearbox is selected Betracht most kommenden enthĂ¤lt. easily byGetriebetypen means ofits a form according to Table 39, which contains all suitable typestype of gearboxes. most easily by means of a form according to Table 39, which contains all suitable types of gearboxes. Das Formular ermĂśglicht die Auswahl 2 Schritten: The form allows selection in 2 in steps: 1. Im ersten werden die Getriebetypen eliminiert, die gearbox Randbedingungen, diecannot von The allows in steps:types In the step, die all the types 1.form In Schritt the firstselection step, allalle thetwo gearbox arefirst eliminated which cannot meet thewhich technical requirements of the application. Technical criteria are used for this purpose. If a der Anwendung vorgegeben werden, nicht erfĂźllenexclusion kĂśnnen. Dazu werden Ausschlusskriterien meet the technical requirements of the application, are eliminated. Technical exclusion criteria are gearbox cannot coverdiese thesenicht criteria, it is excluded second step. zweiten verwendet. Kann type ein Getriebetyp abdecken, wird erfor fĂźrthe den folgenden used for a Table gearbox cannot covercriterion these criteria, it is excluded for the second Inthis the purpose. upper partIfof 39 type for each exclusion it is marked, if the gearbox type Schritt ausgeschlossen. the criterion (table fieldfor marked with an "x"), or if it doesnâ&#x20AC;&#x2122;t (table field empty). Im oberen Teil der Tabelle ist zeilenweise jeden Getriebetyp angegeben, es das type step. Inmeets the upper part of 39 Table 39 each fĂźr exclusion criterion it is marked, if ob theis gearbox 2. InerfĂźllt the second step, the types are compared with each(Tabellenfeld another and the Kriterium (Tabellenfeld ist remaining mit einem gearbox "x" gekennzeichnet) oder nicht erfĂźllt meets the criterion (tableisfield marked Performance with an "x"), criteria or if it doesnâ&#x20AC;&#x2122;t field is empty). optimum solution determined. areGetriebetyp used(table for this. ist leer) bzw. fĂźr welchen Bereich der GetriebeĂźbersetzung der verfĂźgbar ist. In the lower part of Table 39, for each performance criterion it is indicated, how the gearbox type satisfies the criterion (table field is filled with a corresponding performance number). 42 48

The following exclusion criteria are used:

Motion conversion with gears In the second step, the remaining gearbox types are compared with each another and the optimum solution is determined. Performance criteria are used for this. In the lower part of Table 39, for each performance criterion it is indicated, how the gearbox type satisfies the criterion (table field is filled with a corresponding performance number). The following exclusion criteria are used: Gear ratio Each gearbox type is available in a certain gear ratio range. If a larger or smaller gear ratio is required by the application, the gearbox type has to be eliminated. Some gearbox types may have the capability of self-locking or backing in certain ranges of the gear ratio. If these properties are required, the gear ratio valid in this case has to be used as exclusion criterion. Output direction In relation to the driving axis, the driven axis of the gearbox can either be axially or in right angle. Depending on the installation space available in the application, the required output direction must be selected. Shaft type driven side The output shaft can be designed either as a solid shaft or as a hollow shaft. To exclude technically unsuitable gearbox types, select in the upper part of Table 39 in second column the criteria valid for the application. Enter the minimum and the maximum permissible value for the gear ratio. Then, go to the right in each selected row and delete all columns that are not marked with "x" or in which the requirements for the gear ratio are not covered (the corresponding fields are empty). The gearbox types in the deleted columns do not have to be considered in the following second selection step. The following performance criteria are used: Low backlash When the torque changes its direction, only a small angular change occurs without traction between the driving side and the driven side. Low overall length A small overall length in the axial direction leads to a shorter motor-gear combination. Low cross section A small width leads to a gearbox that does not exeed the direction of the motor diameter.

Motion conversion with gears Low axle offset Input and output axes are in line and simplify the machine design. Suitable for S1 operation The gearbox is optimized for S1 operation (continuous operation). Suitable for S5 operation The gearbox is optimized for S5 operation (intermittent operation). Long lifetime The gearbox is designed for high mileages. High efficiency The gearbox is energy efficient. High peak load The gearbox can be heavily overloaded for a short time and is suitable for servo motors. Low moment of inertia A low moment of inertia improves the acceleration capability of the motor. High gear reduction per stage High gear reduction per stage saves stages and improves efficiency. Low noise The gearbox runs quietly.

Motion conversion with gears Low base friction A low friction improves the efficiency and reduces the heating of the gearbox and the motor. Low backing torque The backing torque describes the torque that must be applied to the output shaft in order to keep the motor in rotation at a certain speed. This property is important for door motors in order to open the door manually in case of power loss. Low cost/torque ratio Self-locking effect The gearbox cannot be reset and can therefore be used, for example, for linear actuators without brakes. To select the optimum gearbox type from the technically suitable gearbox types, you have to define the priorities of the performance criteria for the application under consideration in the lower part of Table 39 in the second column. If the criterion is very important to you, set the value to 10. If the criterion is unimportant, enter 0 for the priority. Then, move to the right in each selected row and multiply the priority by the performance number and enter the resulting score into the corresponding field. If, for all technically suitable gearbox types, the scores for all performance criteria are determined, add these columns to a total score. The gearbox type with the highest overall score is the optimum gearbox type that should be selected for the application under consideration.

Planetary gearbox, low backlash

Wave gearbox

Cycloid gearbox

Worm gearbox

Helical gearbox

Needed range of gear ratio

Planetary gearbox

Available gearbox types

Spur gearbox

Motion conversion with gears

Gear ratio Without self-locking or backing 8 22

Minimum gear ratio

Minimum available 200 ratio 200 32 gear

24,5

Minimum gear ratio

Maximum available gear ratio 28

Maximum gear ratio

100

Exclusion criteria

With self-locking

With backing

Minimum gear ratio

Maximum gear ratio

3 3 does not Gearbox type cover exclusion criterion 50 50

Gearbox type covers exclusion criterion x x x

Output direction Axially Right angle

Shaft type driven side Solid shaft

3 20

Selection of the required exclusion criteria Choose one for each criterion.

Hollow shaft

3 15

Suitability Features

Performance criteria

Low backlash

Low overall length

Low cross section

Low axle offset

Suitable for S1 operation

Suitable for S5 operation

2 4 criteria 4 Priority of performance

Long life time

0: not important to 10: 1 3 very important 5

High efficiency

High peak load

Low moment of inertia

High gear reduction per stage

Low noise

4 4 Performance number 5

Low base friction

Low backing torque

Low cost/torque ratio

Ranking

Table 39 Form for selecting the optimum gearbox type

Enter product from priority and performance 4 3 number 5

Enter the sum of the points

Spiroid gearbox

Hypoid gearbox

Bevel gearbox

Crown gearhead

Combination of planetary gearbox and bevel gearbox Combination of planetary gearbox and worm gearbox Combination of spur gearbox and bevel gearbox Combination of planetary gearbox and bevel gearbox for high demands

Combination of planetary gearbox and crown gearhead

Motion conversion with gears

4 10 1 2 4,5 9 3 4,5 18

75 20 5 10 30 30 60 60 60

4 10 1 2 4,5 9 3 4,5 18

15 20 5 10 30 30 60 60 60

x x x x x x x x x

x x

2 4 3 2 2 3 4 4 3

2 2 3 3 2 2 2 3 5

3 4 3 1 3 1 4 5 5

3 4 5 5 5 1 3 5 5

5 5 3 1 3 3 4 5 3

5 5 3 3 3 3 4 5 4

5 5 2 1 2 3 4 4 3

Continuation Table 39

Motion conversion with gears

4.3.4

Selection of the gearbox

The gearbox is selected in two steps. 1. Selection of the gearbox according to mechanical characteristics 4.3.4

Auswahl des Getriebes

2. Selection of of the gearbox according to thermal characteristics 4.3.4 Selection gearbox Die Auslegung desthe Getriebes erfolgt in zwei Schritten The gearbox is selected in two steps. 1. Auswahl des Getriebes nach mechanischen KenngrĂśĂ&#x;en When 1. Selection of the gearbox according to mechanical characteristics 2. Auswahl des Getriebes nach thermischen KenngrĂśĂ&#x;en of the gearbox thermal â&#x20AC;˘ 2. the Selection motion sequences on according the driventoside are characteristics determined according to chapter 4.3.1,

â&#x20AC;˘Sind the gear ratio as defined according to chapter 4.3.2 and When die BewegungsablĂ¤ufe der Abtriebsseite entsprechend Kapitel 4.3.1ermittelt, motion sequences on the driven side are determined according to chapter 4.3.1, â&#x20AC;˘ â&#x20AC;˘theâ&#x20AC;˘ the optimum gearbox type isandetermined according to chapter 4.3.3, â&#x20AC;˘ die GetriebeĂźbersetzung entsprechend Kapitel 4.3.2 festgelegt und â&#x20AC;˘ the gear ratio as defined according to chapter 4.3.2 and the concrete gear is Getriebetyp selected from the catalogue. optimale entsprechend Kapitel to 4.3.3 bestimmt, â&#x20AC;˘ â&#x20AC;˘ theder optimum gearbox type is determined according chapter 4.3.3, wird das purpose, konkrete Getriebe austhe dem Katalog ausgewĂ¤hlt. Dazu werden relevanten Arbeitspunkte this the relevant operating points of the load cycle are die transferred into the charactheFor concrete gear is selected from catalogue. For this purpose, the relevant operating points of the des Lastspiels in das Kennliniendiagramm Getriebes Dabei die Vorzeichen load cycle are transferred into the characteristicdes diagram of theĂźbertragen. gearbox. The signswerden of the torque and teristic diagram of the gearbox. The signs of the torque and the rotational speed are neglected. Drehmoments und der Drehzahl vernachlĂ¤ssigt. thedes rotational speed are neglected. Lastspiel Load cycle Load cycle

Selection condition Auswahlbedingung Selection condition

Kennlinien des Getriebes Characteristics of the gearbox Characteristics of the gearbox

For allFĂźr segments of the alle Abschnitte Forcycle all segments the load applies: of des Lastspiels gilt: load cycle applies:

M2N: Rated torque driven side Nenndrehmoment Abtriebsseite torque driven side M2N: Rated M2max: Maximum permissible torque driven side Maximalpermissible zulĂ¤ssigestorque Drehmoment Abtriebsseite driven side M2max: Maximum n1N: Rated speed driving side Nenndrehzahl Antriebsseite Rated speed driving side n 1N: n1max: Maximum permissible speed driving side Maximum driving side Maximalpermissible zulĂ¤ssige speed Drehzahl Antriebsseite n 40: Selection Table1max of the gearbox according to mechanical characteristics Tabelle 40 Auswahl Getriebes nach mechanischen KenngrĂśĂ&#x;en Table 40 Selection of thedes gearbox according to mechanical characteristics

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If all relevant operating points of the load cycle are located within the area limited by the characteristic alle markanten Arbeitspunkte des Lastspiels innerhalb Kennlinie fĂźr den zyklischen forLiegen cyclic operation, the gearbox is suitable from a mechanical pointder of view. ist das Getriebe unter of mechanischen Gesichtspunkten geeignet. IfBetrieb, all gearboxes relevant operating points thefurther load cycle are located within theselection area limited characteriThese are pre-selected and considered in the following steps.byAllthe other Diese Getriebe werden vorausgewĂ¤hlt und in den folgenden Auslegungsschritten weiter betrachtet. gearboxes are neglected. stic for cyclic operation, the gearbox is suitable from a mechanical point of view. These gearboxes Alle anderen Getriebe werden aussortiert. areenergy pre-selected andwithin further in the following selection All other gearboxes are The conversion theconsidered gearbox is subject to loss, which leadssteps. to temperature rise within the Die Energiewandlung im Getriebe verlustbehaftet, was zur des Getriebes fĂźhrt. Es gearbox. It hasThe to be ensured that the ist heating does not exceed theErwĂ¤rmung permissible limits. neglected. energy conversion within the gearbox is subject to loss, which leads to temperature muss sichergestellt werden, dass die ErwĂ¤rmung die zulĂ¤ssigen Grenzwerte nicht Ăźberschreitet. rise within theoperation gearbox.characteristic It has to be of ensured that the heating does not exceed the permissible The permanent the gearbox indicates the thermally permissible continuous limits. torque as a functionoperation of the des speed. Since variable speed drivesindicates and servo drives have a temporally Die Dauerlastkennlinie Getriebes gibtofdas thermisch zulĂ¤ssige Dauerdrehmoment als Funktion The permanent characteristic the gearbox the thermally permissible contvariable load, thisan. must converted to a continuous equivalent load. The continuous equivalent der Drehzahl Da be drehzahlverĂ¤nderliche Antriebe und Servoantriebe eine zeitlich verĂ¤nderliche inuous torque as gearbox a function of the speed. Since variable and servo motors have a torque heats up the in the same way the torque of speed the realmotors load cycle. Belastung aufweisen, muss diese auf eineas Ersatzdauerlast bzw. einen Ersatzarbeitspunkt The continuous equivalent torque or the equivalent operating point is described the RMS of umgerechnet werden. Die Ersatzdauerlast erwĂ¤rmt das Getriebe in gleicherbyWeise wievalue das reale theLastspiel. torque at the driven side and the mean absolute speed at the driving side. Die Ersatzdauerlast bzw. der Ersatzarbeitspunkt werden durch das Effektivmoment auf der 54 Abtriebsseite und die mittlere absolute Drehzahl auf der Antriebsseite beschrieben.

Motion conversion with gears temporally variable load, this must be converted to a continuous equivalent load. The continuous equivalent torque heats up the gearbox in the same way as the torque of the real load cycle. The continuous equivalent torque or the equivalent operating point is described by the RMS value of the torque at the driven side and the mean absolute speed at the driving side. LoadLastspiel cycle Load cycle

Characteristics ofGetriebes theofgearbox Kennlinien des Characteristics the gearbox

Continuous equivalent load causing same Ersatzdauerlast mitload gleicher ErwĂ¤rmung heating Continuous equivalent causing same heating

M2eff:

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

RMS value of the torque, value of the torque, M2eff : RMS Effektivmoment, allgemeine Definition general definition general definition

M2eff: M RMS value of the of torque, simplified RMS value the torque, simplified Effektivmoment, vereinfachte Formel 2eff: formula segment wise constant formula for segment wise constant fĂźr for abschnittsweise konstantes torque at driven side torque at driven side Drehmoment auf der Abtriebsseite n1av:

Mean absolute speed, Mean absolute speed, n1av: Mittlere absolute Drehzahl, allgemeine general definition general definition Definition

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đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??ź + đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = â&#x2C6;&#x161; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; 2đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

1 đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = 1 â&#x2C6;Ť|đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1 |đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; ďż˝|đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1 |đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0 0

The equivalent operating point is located within the area Der equivalent Ersatzarbeitspunkt liegt unterhalb der Kennlinie The operating point is located within the area Selection condition Selection condition Auswahlbedingung limited by thethe characteristic forfor permanent operation. limited characteristic permanent operation. fĂźr denbyDauerbetrieb. M2 : Torque driven side n1max: Maximum permissible speed driving side Torque driven side Maximum permissible speed (1/s) Drehmoment Abtriebsseite Maximal zulĂ¤ssige M2 : n1max: M2N: Rated torque driven side M2I, 2II, 2III, 2IV: Torque driven side within segment I, II, III, IV Rated torque driven side driving sideAntriebsseite Nenndrehmoment Abtriebsseite Drehzahl M2N: M2max: Maximum permissible torque driven side TI, II, III, IV: Duration of segment I, II, III, IV Maximal zulĂ¤ssiges Drehmoment permissible torque driven side within : Torque Drehmoment Abtriebsseite M2max: Maximum M n1: Speed driving side T: 2I, 2II, 2III, 2IVDuration of load cycle Abtriebsseite im Abschnitt II, III, IV Dauer driven side segment I, II, III,I, IV n1N: Rated speed driving side Drehzahl Antriebsseite des Abschnitts I, II,I,III, IV IV Speed driving side (1/s) Duration of segment II, III, n: TI, II, III, IV: Table 141 Selection of the gearbox according to thermal characteristics Nenndrehzahl Antriebsseite Dauer des Lastspiels Rated speed driving side (1/s) Duration of load cycle n1N: T: Tabelle 41 Auswahl Getriebes KenngrĂśĂ&#x;en If the equivalent operatingdes point is locatednach withinthermischen the area limited by the characteristic for permanent Table 41 Selection of the gearbox according to thermal characteristics

operation, the corresponding gearbox is suitable from a thermal point of view.

Lieg der Ersatzarbeitspunkt unterhalb der Kennlinie fĂźrTable den Dauerbetrieb, ist gearbox das entsprechende Of all gearboxes meeting the selection condition shown in 41, the smallest is selected. If the equivalent operatingGesichtspunkten point is locatedgeeignet. within the area limited by the characteristic for Getriebe unter thermischen Afterpermanent selecting the gearbox, beside the speed n1 the torque is profile M1 atfrom the driving side can beof view the operation, the corresponding gearbox suitable a thermal point Von allen Getrieben, die die41. Auswahlbedingung 41 erfĂźllen, wird das the kleinste determined according to Table These two profilesentsprechend are importantTabelle input variables for selecting corresponding gearbox is suitable. Getriebe ausgewĂ¤hlt. motor.

Nach Auswahl des Getriebes kann entsprechend der Tabelle 37 neben dem Drehzahlverlauf n1 auch der Drehmomentverlauf M1 an der Antriebsseite ermittelt werden. Diese beiden GrĂśĂ&#x;en sind wichtige Parameter zur Auswahl des Motors.

Motion conversion with gears Of all gearboxes meeting the selection condition shown in Table 41, the smallest gearbox is selected. After selecting the gearbox, beside the speed n1 the torque profile M1 at the driving side can be determined according to Table 41. These two profiles are important input variables for selecting the motor.

4.3.5 Gearboxes from Dunkermotoren 4.3.5.1 Planetary Gearboxes - Series PLG

Planetary gearboxes offer the highest continuous torque capacity of all types of gearboxes. The design is very compact, the weight is low and the efficiency is excellent. For planetary gearboxes from Dunkermotoren there are depending on customer requirements high-power, low-noise or low-backlash versions available. Continuous torques reach up to 160 Nm and the ratios vary from 3:1 to 710.5:1.

4.3.5.2 Spirotec Gearboxes - Series STG

Spirotec gearboxes are gearboxes with right angle output. Core element of the series STG is the spiral wheel set. It enables reliable transmission of high torque with comparatively small centre distance in a small package. The Spirotec Gearbox is outstandingly quiet in operation and provides an unmatched long lifetime.

Planetengetriebe haben die hĂśchsten zulĂ¤ssigen Dauerdrehmomente aller Getriebe bei gleichzeitig sehr kompakter Bauform, geringem Gewicht und ausgezeichnetem Wirkungsgrad. Bei unseren Planetengetrieben ist eine Vielzahl von verschiedensten Materialien je nach Kundenanforderungen kombinierbar, von- verstĂ¤rkten 4.3.5.3 Worm/helical Gearboxes Series SGĂźber besonders laufruhige bis hin zu spielarmen Varianten. Es sind Dauerdrehmomente bis zu 160 Nm mĂśglich und Untersetzungen von 4:1 bis 710,5:1 erhĂ¤ltlich.

Motion conversion with gears

Worm/helical gearboxes are gearboxes with 4.3.5.2 Spirotecgetriebe - Baureihe STG Spirotecgetriebe sindand Getriebe Abtrieb. Das right angle output notedmit forrechtwinkligem their very HerzstĂźck der Baureihe STG ist der spiralverzahnte Radsatz. quiet running. The worm gear shaft has Dieser ermĂśglicht es mit vergleichsweise geringem bearings on auf both sides Bauraum and is available with zuverlĂ¤ssig Achsabstand kleinem hohe Momente zu Ăźbertragen. Spirotecgetriebe different shaftDas positions or hollow zeichnet shaft. sich besonders hohe Laufruhe und eine herausragende Lebensdauer aus. 4.3.5.3 Schneckengetriebe - Baureihe SG Schneckengetriebe sind Getriebe mit rechtwinkligem Abtrieb und zeichnen sich durch hohe Laufruhe aus. Die 4.3.5.4 Bevel Gearboxes - Series KG Schneckenradwelle ist beidseitig gelagert und ist in verschiedenen Wellenlagen bzw. mit Hohlwelle erhĂ¤ltlich.

Bevel gearboxes are gearboxes with zero-offset right angle output. They have a high efficiency and allow in combination 4.3.5.4 Kegelradgetriebe - Baureihe with planetary stages a wide range ofKG gear Kegelradgetriebe sind Getriebe mit ratios. rechtwinkligem Abtrieb ohne Achsversatz zur Motorwelle. Sie zeichnen sich durch einen hohen Wirkungsgrad sowie eine groĂ&#x;e Bandbreite mĂśglicher Untersetzungen in Kombination mit Planetenstufen aus.

4.4

Example: Pump motor

After plant has been configured and the pump has been selected, the required characteris4.4 the Beispiel: Pumpenantrieb Nachdem die Anlage projektiert und die Pumpe ausgewĂ¤hlt wurde, stehen die erforderlichen tics are available. They are entered into a common diagram. Kennlinien zur VerfĂźgung. Sie werden in ein gemeinsames Diagramm eingetragen. Daten Pumpe fĂźratn n= = 2600 1/min Datader of the pump 2600 rpm Q in mÂł/h

0,00

0,32

0,48

0,64

0,80

0,96

1,12

1,28

1,36

1,54

H in m

11,00

10,25

9,70

9,00

8,20

7,10

6,00

4,95

4,35

2,75

đ?&#x203A;Ľđ?&#x203A;Ľđ?&#x203A;Ľđ?&#x203A;Ľđ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ??ťđ??ťđ??ťđ??ť â&#x2C6;&#x2122; đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152;đ?&#x153;&#x152; â&#x2C6;&#x2122; đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; in kN/mÂ˛

107,9

100,6

95,2

88,3

80,4

69,7

58,9

48,6

42,7

27,0

Îˇ

0,00

0,32

0,45

0,56

0,64

0,66

0,65

0,62

0,58

0,41

PP in W

0,0

8,9

12,7

15,7

17,9

18,6

18,3

17,3

16,1

11,5

49,1

49,9

50,9

52,3

54,1

56,4

59,0

62,1

63,7

67,9

Data of the plant Daten der Anlage đ?&#x203A;Ľđ?&#x203A;Ľđ?&#x203A;Ľđ?&#x203A;Ľđ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´ in N/mÂ˛

Motion conversion with gears Kennlinien der Pumpe Characteristics of the pump

ThestationĂ¤re steady-state operating point is sich givenzu: by: Der Arbeitspunkt ergibt

Volumenstrom Volume flow Pressure Druckdifferenz difference

đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 1,12

Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 58,9

Wirkungsgrad Efficiency Conveying FĂśrderleistung capacity

Leistung Antriebsseite, Motorleistung Power at driving side, needed motor power

Drehmoment Antriebsseite, Torque at driving side, needed Motordrehmoment motor torque

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; =

đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;3 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;3 = 0,0003 â&#x201E;&#x17D; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 0,65

đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ â&#x2C6;&#x2122; Î&#x201D;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 17,7 đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ 17,7 đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; = = 27,2 đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ 0,65

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 27,2 đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; â&#x2C6;&#x2122; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = = 0,1 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; 2600

Volumenstrom zu fĂśrdernden Mediums Q: Volume flow of thedes heating water to be pumped FĂśrderhĂśhe der H: Discharge head of Pumpe the pump Dichteofdes zu fĂśrdernden Heizwassers Ď : Density heating water to be pumped (1000 kg/mÂł) (1000 kg/mÂł) Erdbeschleunigung (9,81 m/sÂ˛) g: Acceleration due to gravity Druckdifferenz derofPumpe (Pumpenkennlinie) fĂźr nat=n=2800 2800 1/min difference the pump (pump characteristic) rpm â&#x2C6;&#x2020;pP: Pressure Wirkungsgradkennlinie (Pumpenkennlinie) = 2800rpm 1/min characteristic ofder thePumpe pump (pump characteristic)fĂźr at n n=2800 Îˇ: Efficiency FĂśrderleistung derofPumpe (Pumpenkennlinie) fĂźr n at = n=2800 2800 1/min capacity the pump (pump characteristic) rpm PP: Conveying Druckdifferenz derofAnlage fĂźratnn=2800 = 2800 rpm 1/min difference the plant(Anlagenkennlinie) (plant characteristic) â&#x2C6;&#x2020;pA: Pressure Tabelle 42 Arbeitspunkt eines Pumpenantriebs

Table 42 Operating point of a pump motor

Der Schnittpunkt der Pumpen- und der Anlagenkennlinie liefert den stationĂ¤ren Arbeitspunkt. FĂźr den Pumpenantrieb kommt kein Getriebe zum Einsatz. Die BewegungsgrĂśĂ&#x;en an der Welle der The point of identisch intersection of the pump and the system characteristic provides the steady-state Pumpe sind mit den BewegungsgrĂśĂ&#x;en an der Motorwelle.

operating point. No gearbox is used for the pump motor. The motion variables at the driven shaft the pump are identical to the motion variables at the driving motor shaft. 4.5 ofBeispiel: TĂźrantrieb Das mechanische System des TĂźrantriebs besteht aus einem Seilzug, der die FahrstuhltĂźr bewegt. Es sind folgende Parameter gegeben:

4.5

Example: Door motor

90 kg â&#x20AC;˘ Masse der TĂźr inklusive Halterung mL: 1,5 kg â&#x20AC;˘ Masse des Seils mSZ: The mechanical system of the door motor consists of a wire rope pulling the elevator door. The 0,06 m â&#x20AC;˘ Durchmesser der Umlenkrollen dU: following parameters are given: 0,0002 kgmÂ˛ â&#x20AC;˘ TrĂ¤gheitsmoment einer Umlenkrolle JU: 0,04 m â&#x20AC;˘ Durchmesser der StĂźtzrollen dS: 0,00004 kgmÂ˛ â&#x20AC;˘ TrĂ¤gheitsmoment einer StĂźtzrolle JS: Massâ&#x20AC;˘ of Masse door including bracketmm : 90 kg â&#x20AC;˘ Mass of the wire rope mSZ:0,2 1,5kg kg â&#x20AC;˘ Diameter of the einer StĂźtzrolle S: L â&#x20AC;˘ Anzahl derdU StĂźtzrollen deflection pulleys : 0,06 m â&#x20AC;˘nMoment of inertia of a deflection pulley 2JU: 0,0002 kgmÂ˛ â&#x20AC;˘ Diameter S: 0,0010 kgmÂ˛ â&#x20AC;˘ Mass of a Rollwiderstandskoeffizient c R: of theâ&#x20AC;˘ support rolls dS: 0,04 m â&#x20AC;˘ Moment of inertia of a support roll JS: 0,00004 0,95 â&#x20AC;˘ Wirkungsgrad des Seilzugs Îˇ: support mS: 0,2 kg â&#x20AC;˘ Number ofdes support rolls coefficient 1/min cR: 0,0010 â&#x20AC;˘ â&#x20AC;˘ roll Angestrebte Nenndrehzahl Motors nN: nS: 2 â&#x20AC;˘ Rolling friction 3000

Efficiency of the wire rope Îˇ: 0,95 â&#x20AC;˘ Desired rated speed of the motor nN: 3000 1/min Gemeinsam mit den BewegungsgrĂśĂ&#x;en aus Kapitel 3.6 lassen sich die BewegungsgrĂśĂ&#x;en an der Antriebsseite des Seilzugs ermitteln. The motion variables at the driving side of the wire rope can be determined using the motion variables calculated in chapter 3.6.

Motion conversion with gears Bewegungsgrößen des Seilzugs Motion variables an at der the Antriebsseite driving side of the wire rope

Segment I, Beschleunigen Segment I, acceleration Maximale angular WinkelMaximum geschwindigkeit speed

𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙

Maximale speed Drehzahl Maximum

𝑛𝑛𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙

WinkelAngular beschleunigung acceleration BeschleunigungsAcceleration force kraft Last, of theder load, the des Seils undsupder rope and the port rolls Stützrollen Friction force Reibkraft der of the load Last (v (v >> 0) 0) Linear LineareForce Kraft

Drehmoment Torque driving side, Antriebsseite, motor operation motorischer Betrieb (F > 0, v > 0) (F > 0, v > 0)

2 𝑚𝑚𝑚𝑚 2 1 = 0,4 ∙ = 13,33 𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠 0,06𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠

1 𝑚𝑚𝑚𝑚 1 1 = 0,4 ∙ = 127,32 𝜋𝜋𝜋𝜋𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 ∙ 0,06𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛

𝛼𝛼𝛼𝛼 = 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙

2 𝑚𝑚𝑚𝑚 2 1 = 0,5 ∙ = 16,67 𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠² 0,06𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠²

𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 = �𝑚𝑚𝑚𝑚𝐿𝐿𝐿𝐿 + 𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 + 𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆 + 𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝐽𝐽𝐽𝐽𝑆𝑆𝑆𝑆

𝑑𝑑𝑑𝑑𝑆𝑆𝑆𝑆 2

𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 = �90 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 1,5 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 2 ∙ 0,2 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 2 ∙ 0,00004 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑚𝑚𝑚𝑚²

� ∙ 𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

4 𝑚𝑚𝑚𝑚 � ∙ 0,5 = 46,05 𝑁𝑁𝑁𝑁 (0,04 𝑚𝑚𝑚𝑚)2 𝑠𝑠𝑠𝑠²

𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 𝑐𝑐𝑐𝑐𝑅𝑅𝑅𝑅 ∙ [𝑚𝑚𝑚𝑚𝐿𝐿𝐿𝐿 + 𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆 ] ∙ 𝑘𝑘𝑘𝑘 = 0,001 ∙ [90 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 2 ∙ 0,2 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘] ∙ 9,81 𝐹𝐹𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 + 𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 46,05 𝑁𝑁𝑁𝑁 + 0,89 𝑁𝑁𝑁𝑁 = 46,94 𝑁𝑁𝑁𝑁 𝑀𝑀𝑀𝑀 = 2 ∙ 𝐽𝐽𝐽𝐽𝑈𝑈𝑈𝑈 ∙ 𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + �𝐹𝐹𝐹𝐹 ∙

𝑀𝑀𝑀𝑀 = 2 ∙ 0,0002𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑚𝑚𝑚𝑚² ∙ 16,67

Segment II, II, Konstantfahrt Segment constant speed WinkelAngular speed geschwindigkeit

𝑛𝑛𝑛𝑛 = 𝑛𝑛𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 127,32

WinkelAngular beschleunigung acceleration

𝑎𝑎𝑎𝑎 = 0

BeschleunigungsAcceleration force kraft Last, of theder load, the des Seils undsupder rope and the Stützrollen port rolls

𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 1 �∙ 2 𝜂𝜂𝜂𝜂

1 0,06 𝑚𝑚𝑚𝑚 1 + �46,94 𝑁𝑁𝑁𝑁 ∙ �∙ = 1,49 𝑁𝑁𝑁𝑁𝑚𝑚𝑚𝑚 2 0,95 𝑠𝑠𝑠𝑠²

𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 13,33

Drehzahl Speed

𝑚𝑚𝑚𝑚 = 0,89 𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠²

1 𝑠𝑠𝑠𝑠²

1 𝑠𝑠𝑠𝑠

1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛

𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 = 0 N 50

Motion conversion with gears Reibkraft der of Last Friction force the (v > 0) load (v > 0) LineareForce Kraft Linear

𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 𝑐𝑐𝑐𝑐𝑅𝑅𝑅𝑅 ∙ [𝑚𝑚𝑚𝑚𝐿𝐿𝐿𝐿 + 𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆 ] ∙ 𝑔𝑔𝑔𝑔 = 0,001 ∙ [90 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔 + 2 ∙ 0,2 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔] ∙ 9,81 𝐹𝐹𝐹𝐹 = 𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 0 𝑁𝑁𝑁𝑁 + 0,89 𝑁𝑁𝑁𝑁 = 0,89 𝑁𝑁𝑁𝑁

Drehmoment Torque driving side, Antriebsseite, motor operation motorischer Betrieb (F > 0, v > 0) (F > 0, v > 0)

𝑀𝑀𝑀𝑀 = �𝐹𝐹𝐹𝐹 ∙

𝑚𝑚𝑚𝑚 = 0,89 𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠²

𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 1 0,06 𝑚𝑚𝑚𝑚 1 � ∙ = �0,89 𝑁𝑁𝑁𝑁 ∙ �∙ = 0,03 𝑁𝑁𝑁𝑁𝑚𝑚𝑚𝑚 2 𝜂𝜂𝜂𝜂 2 0,95

Segment III, Segment III,Bremsen deceleration Maximale angular WinkelMaximum geschwindigkeit speed Maximale speed Drehzahl Maximum

Angular Winkelbeschleunigung acceleration BeschleunigungsAcceleration force kraft Last, of theder load, the des Seils undsupder rope and the Stützrollen port rolls

Friction force the Reibkraft der of Last load (v > 0) (v > 0) LineareForce Kraft Linear Drehmoment Torque driving side, Antriebsseite, generator operation generatorischer (F < 0, v > 0) Betrieb (F < 0, v > 0) Segment IV, Segment IV, Pause pause

𝜔𝜔𝜔𝜔 = 𝜔𝜔𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙ 𝑛𝑛𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑣𝑣𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙

1 𝑚𝑚𝑚𝑚 1 1 = 0,4 ∙ = 127,32 𝜋𝜋𝜋𝜋𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋 ∙ 0,06𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛

𝛼𝛼𝛼𝛼 = −𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = −𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∙

2 𝑚𝑚𝑚𝑚 2 1 = −0,5 ∙ = −16,67 𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠² 0,06𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠²

𝑑𝑑𝑑𝑑𝑆𝑆𝑆𝑆 2

𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 = �90 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔 + 1,5𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔 + 2 ∙ 0,2𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔 + 2 ∙ 0,00004 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔²

� ∙ (−𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )

4 𝑚𝑚𝑚𝑚 � ∙ (−0,5) = −46,05 𝑁𝑁𝑁𝑁 (0,04 𝑚𝑚𝑚𝑚)2 𝑠𝑠𝑠𝑠²

𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 𝑐𝑐𝑐𝑐𝑅𝑅𝑅𝑅 ∙ [𝑚𝑚𝑚𝑚𝐿𝐿𝐿𝐿 + 𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆 ] ∙ 𝑔𝑔𝑔𝑔 = 0,001 ∙ [90 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔 + 2 ∙ 0,2 𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔] ∙ 9,81 𝐹𝐹𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 + 𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = −46,05 𝑁𝑁𝑁𝑁 + 0,89 𝑁𝑁𝑁𝑁 = −45,14 𝑁𝑁𝑁𝑁 𝑀𝑀𝑀𝑀 = 2 ∙ 𝐽𝐽𝐽𝐽𝑈𝑈𝑈𝑈 ∙ (−𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) + �𝐹𝐹𝐹𝐹 ∙

𝑀𝑀𝑀𝑀 = 2 ∙ 0,0002𝑘𝑘𝑘𝑘𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔² ∙ �−16,67

WinkelAngular speed geschwindigkeit

𝑛𝑛𝑛𝑛 = 0

Angular Winkelbeschleunigung acceleration

1 𝑠𝑠𝑠𝑠

1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛

𝑎𝑎𝑎𝑎 = 0

BeschleunigungsAcceleration Force kraft der Last, of the load, the rope des the Seils und der and support rolls Stützrollen

1 𝑠𝑠𝑠𝑠²

𝐹𝐹𝐹𝐹𝐵𝐵𝐵𝐵 = 0 N

Reibkraft der of Last Friction force the (v = 0) load (v > 0)

𝐹𝐹𝐹𝐹𝑅𝑅𝑅𝑅 = 0 𝑁𝑁𝑁𝑁

LineareForce Kraft Linear

Table 43 Motion variables of the wire rope for a door motor 51

𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 � ∙ 𝜂𝜂𝜂𝜂 2

𝑚𝑚𝑚𝑚 = 0,89 𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠²

1 0,06 𝑚𝑚𝑚𝑚 � + �−45,14 𝑁𝑁𝑁𝑁 ∙ � ∙ 0,95 = −1,29 𝑁𝑁𝑁𝑁𝑚𝑚𝑚𝑚 2 𝑠𝑠𝑠𝑠²

𝜔𝜔𝜔𝜔 = 0

Drehzahl Speed

2 𝑚𝑚𝑚𝑚 2 1 = 0,4 ∙ = 13,33 𝑑𝑑𝑑𝑑𝑈𝑈𝑈𝑈 𝑠𝑠𝑠𝑠 0,06𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠

𝐹𝐹𝐹𝐹 = 0 𝑁𝑁𝑁𝑁

Motion conversion with gears The motion variables for the driving side of the wire rope do not match the typical characteristics of Diean BewegungsgrĂśĂ&#x;en an der a Antriebsseite des Seilzugs passen nicht zurange den typischen electric motor. Therefore, rotating gearbox is required. The possible of the gearKenngrĂśĂ&#x;en ratio must eines Elektromotors. Deshalb ist ein rotatorisches Getriebe erforderlich. determined first. According to the task, maximum speed at zu thebestimmen. driving side Laut should be less than Alsbe erstes ist der mĂśgliche Wertebereich derthe GetriebeĂźbersetzung Aufgabenstellung soll die maximale Drehzahl auf der Antriebsseite unterhalb von 3000 1/min liegen. 3000 rpm. This results in a maximum value for the gear ratio of Damit ergibt sich fĂźr die GetriebeĂźbersetzung ein Maximalwert von đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; =

3000 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = 1 = 23,56 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;2đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 127,32 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

Mit dieser Angabe kann der optimale Getriebetyp ermittelt werden. Dazu werden im ersten Schritt dieWith technisch geeigneten Getriebetypen Entsprechend Tabelle 46For kommen die folgenden this specification, the optimum extrahiert. gearbox type can be determined. this purpose in the Getriebetypen in Betracht:

first step the technically suitable gearbox types are extracted. According to Table 46, the folloâ&#x20AC;˘ Schraubradgetriebe wing gearbox types are suitable: â&#x20AC;˘ Helical gearbox â&#x20AC;˘ Hypoid gearbox â&#x20AC;˘ Combination of planetary â&#x20AC;˘

Hypoidgetriebe

gearbox and bevelaus gearbox â&#x20AC;˘ Combination planetary gearbox and worm gearbox â&#x20AC;˘ Combinaâ&#x20AC;˘ Kombination Planetengetriebe und of Kegelradgetriebe â&#x20AC;˘ Kombination aus Planetengetriebe und Schneckengetriebe tion of spur gearbox and bevel gearbox â&#x20AC;˘ Combination of planetary gearbox and bevel gearbox â&#x20AC;˘

Kombination aus Stirnradgetriebe und Kegelradgetriebe

for demands.aus ForPlanetengetriebe the second step,und theKegelradgetriebe requirements of the application are weighted and â&#x20AC;˘ high Kombination fĂźr hohe Anforderungen the priorities are assigned:

Im zweiten Schritt werden die Anforderungen der Anwendung gewichtet und priorisiert: Leistungskriterium

Performance criterion

Prio-

Priority ritĂ¤t

BegrĂźndung

Reason

Keine hohe PrioritĂ¤t, da keine sehr hohe Genauigkeit gefordert Geringes Umkehrspiel 2 Low backlash 2 ist.No high priority, since very high accuracy is not required. Keine hohe PrioritĂ¤t , da ein rechtwinkliges Getriebe verwendet Geringe BaulĂ¤nge 2 wird. Low overall length 2 No high priority, since a right-angle gearbox is used. Geringer Bauquerschnitt Keine hohe PrioritĂ¤t , da ein rechtwinkliges Getriebe verwendet 2 wird. No high priority, since a right-angle gearbox is used. Low cross section 2 Geringer Achsversatz 0 Der Achsversatz spielt keine Rolle. FĂźrLow S1-Betrieb geeignet 00 Dauerbetrieb ist nicht axle offset The axle offset is notnotwendig. important. FĂźr S5-Betrieb geeignet 1 Der Aussetzbetrieb ist erforderlich, hat aber keine hohe PrioritĂ¤t. Hohe Lebensdauer 50 DerContinuous Antrieb wirdoperation nur sporadisch Suitable for S1 operation is not benutzt. needed. Hoher Wirkungsgrad 5 Eine hohe Energieeffizienz ist erwĂźnscht. Intermittent operation is required but is not a high Das Getriebe wird nicht mit sehr hohen Drehmomentspitzen Suitable for S5 operation Hohe Spitzenlast 21 priority. belastet. DieThe EigentrĂ¤gheit desused Getriebes wird aufgrund der hohen Masse Long lifetime motor is only sporadically. Geringes TrĂ¤gheitsmoment 15 der TĂźr kaum eine Rolle spielen. Hohe Untersetzung 55 Eine hohe Untersetzung verbessert den Wirkungsgrad. High efficiency je Stufe High efficiency is desired. Da der Aufzug im Wohnbereich eingesetzt wird, ist eine geringe Geringe GerĂ¤usche 10 GerĂ¤uschentwicklung sehr wichtig. The gearbox is not exposed to very high torque peaks. High peak load 2 Geringe Grundreibung 2 Die Grundreibung spielt eine geringe Rolle. Das RĂźcktreibmoment spielt bei einem TĂźrantrieb The inherent moment of inertia of the gearboxeine has wichtige Geringes RĂźcktreibmoment 10 Low moment of inertia 1 Rolle. a low influence due to the high mass of the door. Niedriges Kosten/Drehmoment10 Da es sich um einen Serieneinsatz handelt, sind niedrige Kosten High gear reduction reductionsehr per stage improves efficiency. 5 VerhĂ¤ltnis fĂźr High das Getriebe wichtig. per stage Tabelle 44 Priorisierung der Leistungskriterien des Getriebes fĂźr einen TĂźrantrieb

Low noise

Low base friction

Since the elevator is used in the living area, a low

noise level is very important. Damit ergeben sich die folgenden Punktezahlen fĂźr die geeigneten Getriebetypen:

The basic friction plays a minor role.

Getriebetyp Punkte The backing torque plays an important role for198 a Schraubradgetriebe Low backing torque 10 Hypoidgetriebe 183 door motor. Kombination aus Planetengetriebe und Kegelradgetriebe Since it is a series application, low cost for the166 gear Low cost/torque ratio 10Schneckengetriebe Kombination aus Planetengetriebe und 143 box is very important. Kombination aus Stirnradgetriebe und Kegelradgetriebe 175 Table 44 Prioritisation of the performance for selecting the gearbox type for a door motor168 Kombination aus Planetengetriebe undcriteria Kegelradgetriebe fĂźroptimum hohe Anforderungen

Tabelle 45 Ranking der Getriebetypen fĂźr einen TĂźrantrieb Der optimale Getriebetyp ist damit ein Schraubradgetriebe. 51

Motion conversion with gears This results in the following scores for the suitable gearbox types: Gearbox type Helical gearbox

198

Hypoid gearbox

183

Combination of planetary gearbox and bevel gearbox

166

Combination of planetary gearbox and worm gearbox

143

Combination of spur gearbox and bevel gearbox

175

Combination

168

Table 45 Ranking of gearbox types for a door motor

The optimum gearbox type is thus a helical gearbox.

Score

Spur gearbox

Planetary gearbox

Planetary gearbox, low backlash

Wave gearbox

Cycloid gearbox

Worm gearbox

Helical gearbox

Motion conversion with gears

Minimum gear ratio

Maximum gear ratio

100

200

24,5

Gear ratio Without self-locking or backing

Exclusion criteria

With self-locking Minimum gear ratio

Maximum gear ratio

With backing 15

Minimum gear ratio

Maximum gear ratio

Output direction Axially

Right angle

Shaft type driven side x

Solid shaft

Hollow shaft

Suitability

Performance criteria

Features 2

Low backlash

Low overall length

Low cross section

Low axle offset

Suitable for S1 operation

Suitable for S5 operation

Long life time

High efficiency

High peak load

Low moment of inertia

High gear reduction per stage

Low noise

Low base friction

Low backing torque

Low cost/torque ratio

10 2

Ranking

198

Table 46 Form for selecting the optimum gearbox type for a door motor

Spiroid gearbox

Hypoid gearbox

Bevel gearbox

Crown gearhead

Combination of planetary gearbox and bevel gearbox

Combination of planetary gearbox and worm gearbox

Combination of spur gearbox and bevel gearbox

Combination of planetary gearbox and bevel gearbox for high demands

Combination of planetary gearbox and crown gearhead

Motion conversion with gears

4,5

20 75

4,5

166

143

175

168

183 Continuation Table 46

Motion conversion with gears For the preselection of the gearbox, the maximum occurring torque and the RMS value of the Zur Vorauswahl des Getriebes wird das maximal auftretende Drehmoment sowie das Effektivmoment torque at the drivenbetrachtet. side are considered. auf der Abtriebsseite Maximales

Maximum torque Drehmoment driven side

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = 1,49 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Abtriebsseite Zur Vorauswahl des Getriebes wird das maximal auftretende Drehmoment sowie das Effektivmoment auf der Abtriebsseite betrachtet. (1,49 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,03 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;1,29 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Effektivmoment RMS value of the đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ = 0,612 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; torque driven side Abtriebsseite 8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Maximales Drehmoment đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = 1,49 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Abtriebsseite Mit diesen kann values, ein Getriebe aus dercan nachfolgend angegebenen With theseKennwerten characteristic a gearbox be selected from the following worm or helical Schraubradgetriebebaureihe ausgewĂ¤hlt werden. gearbox series. (1,49 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,03 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;1,29 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Effektivmoment đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ = 0,612 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Abtriebsseite 8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Data/ Technische Daten | SG 80 | SG 80 H | SG 80 K Reduction ratio/ UntersetzungsverhĂ¤ltnis

Efficiency/ Wirkungsgrad

SG 80 / SG 80 H

350

400

Mit diesen Kennwerten kann ein Getriebe aus der nachfolgend angegebenen Continuous torque/ Dauerdrehmoment Ncm 200 250 350 Schraubradgetriebebaureihe ausgewĂ¤hlt werden. Max.acceleration torque/ Max. Beschleunigungsmoment

Emergency torque/ Not-Aus Drehmoment Operating mode/ Betriebsart Reduction ratio/ UntersetzungsverhĂ¤ltnis Efficiency/ Wirkungsgrad Continuous torque/ Dauerdrehmoment

Ncm Ncm

1200

S1 / S5*

SG 80 K

24.5

Ncm

250

350

700

Max.acceleration 9 torque/ Abbildung Auszug aus einem Katalog fĂźr Ncm Schraubradgetriebe 500 500 Max. Beschleunigungsmoment Emergency torque/ Not-Aus Drehmoment

800

Ncm

1200

Das Getriebe weist ein Nenndrehmoment von 3,5 Nm und ein zulĂ¤ssiges Maximaldrehmoment von Operating mode/ Betriebsart S5* 7,0 Nm auf. Damit gilt: Weight of gearbox/ Getriebegewicht

0.9

Axial load / radial load/ Axiallast / Radiallast M2eff < MN

0,612 Nm < 3,5 Nm

300 / 350

Figure 9 Excerpt from a catalogue for helical gearboxes

M2max < Mmax 1,49 Nm < 7,0 Nm Abbildung 9 Auszug aus einem Katalog fĂźr Schraubradgetriebe The gearbox has a rated torque of 7.0 Nm. i < imaxtorque of 3.5 Nm and a permissible maximum 15 < 23,56 Das Getriebe weist ein Nenndrehmoment von 3,5 Nm und ein zulĂ¤ssiges Maximaldrehmoment von Therefore: 7,0 Nm auf. Damit gilt: im zulĂ¤ssigen Bereich. Das ausgewĂ¤hlte Getriebe ist geeignet und kann Alle Kennwerte liegen verwendet werden. Das Getriebe weist ein TrĂ¤gheitsmoment J von 0,0001 kgmÂ˛ auf. 0,612 Nm < 3,5 Nm M2eff < MN Gemeinsam mit den BewegungsgrĂśĂ&#x;en aus Tabelle 43 lassen sich die BewegungsgrĂśĂ&#x;en an der Antriebsseite des Getriebes M2max < Mermitteln. 1,49 Nm < 7,0 Nm max

i < imax BewegungsgrĂśĂ&#x;en an der Antriebsseite des Getriebes

15 < 23,56

Alle Kennwerte liegen im zulĂ¤ssigen Bereich. Das ausgewĂ¤hlte Getriebe ist geeignet und kann All characteristic are within permissible range. selected gearbox is suitable and verwendet werden. values Das Getriebe weist the ein TrĂ¤gheitsmoment J vonThe 0,0001 kgmÂ˛ auf.

can be used. The moment of inertia "J" of the gearbox is 0.0001 kgmÂ˛. Gemeinsam mit den BewegungsgrĂśĂ&#x;en aus Tabelle 43 lassen sich die BewegungsgrĂśĂ&#x;en an der Together with the motion ermitteln. variables from Table 43, the motion variables at the driving side of the Antriebsseite des Getriebes gearbox can be determined. BewegungsgrĂśĂ&#x;en an der Antriebsseite des Getriebes

verwendet werden. Das Getriebe weist ein TrĂ¤gheitsmoment J von 0,0001 kgmÂ˛ auf. Gemeinsam mit denwith BewegungsgrĂśĂ&#x;en aus Tabelle 43 lassen sich die BewegungsgrĂśĂ&#x;en an der Motion conversion gears Antriebsseite des Getriebes ermitteln.

BewegungsgrĂśĂ&#x;en der Antriebsseite des Getriebes Motion variables an at the driving side of the wire rope

Segment I, Beschleunigen Segment I, acceleration 55

Maximale angular WinkelMaximum geschwindigkeit speed

Maximale speed Drehzahl Maximum

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;2 â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; = 127,32

WinkelAngular beschleunigung acceleration Drehmoment Torque driving side, Antriebsseite, motor operation motorischer Betrieb (M2 > 0, Ď&#x2030;2> 0)

1 1 â&#x2C6;&#x2122; 15 = 200 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

1 1 â&#x2C6;&#x2122; 15 = 1910 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź1 = đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;2 â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; = 16,67 đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1 = đ??˝đ??˝đ??˝đ??˝ đ??˝đ??˝đ??˝đ??˝đ??˝1 +

1 1 â&#x2C6;&#x2122; 15 = 250 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2 1 1 1,49 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 1 â&#x2C6;&#x2122; = 0,0001 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ â&#x2C6;&#x2122; 250 + â&#x2C6;&#x2122; = 0,167 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 15 0,7 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

Segment II, Segment II, Konstantfahrt constant speed WinkelAngular speed geschwindigkeit

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;1 = đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = 200

Speed Drehzahl

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1 = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = 1910

WinkelAngular beschleunigung acceleration

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź1 = 0

Drehmoment Torque driving side, Antriebsseite, motor operation motorischer Betrieb

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1 =

(M2 > 0, Ď&#x2030;2> 0)

Segment III, Segment III,Bremsen deceleration Maximale angular WinkelMaximum geschwindigkeit speed Maximale speed Drehzahl Maximum WinkelAngular beschleunigung acceleration Drehmoment Antriebsseite, Torque driving side, generatorischer generator operation Betrieb (M2 < 0, Ď&#x2030;2> 0)

1 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

Winkelgeschwindigkeit

1 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2 1 0,03 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 1 â&#x2C6;&#x2122; = â&#x2C6;&#x2122; = 0,0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; 15 0,7

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;2 â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; = 13,33 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;2 â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; = 127,32

1 1 â&#x2C6;&#x2122; 15 = 200 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź1 = â&#x2C6;&#x2019;đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;2 â&#x2C6;&#x2122; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; = â&#x2C6;&#x2019;16,67 đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1 = đ??˝đ??˝đ??˝đ??˝ đ??˝ (â&#x2C6;&#x2019;đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 ) +

1 1 â&#x2C6;&#x2122; 15 = â&#x2C6;&#x2019;250 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2 1 1,29 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; = 0,0001 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ â&#x2C6;&#x2122; ďż˝â&#x2C6;&#x2019;250 ďż˝ â&#x2C6;&#x2019; â&#x2C6;&#x2122; 0,7 = â&#x2C6;&#x2019;0,085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; 15 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

Segment IV, Pause

1 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;1 = 0

1 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Drehmoment Antriebsseite, generatorischer Betrieb (M2 < 0, Ď&#x2030;2> 0)

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1 = đ??˝đ??˝đ??˝đ??˝ đ??˝ (â&#x2C6;&#x2019;đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;1 ) +

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;2 1 1,29 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2C6;&#x2122; đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A;đ?&#x153;&#x201A; = 0,0001 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;Â˛ â&#x2C6;&#x2122; ďż˝â&#x2C6;&#x2019;250 ďż˝ â&#x2C6;&#x2019;Motion conversion â&#x2C6;&#x2122; 0,7 = â&#x2C6;&#x2019;0,085 with đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; gears đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; 15 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

Segment IV, Segment IV, Pause pause WinkelAngular speed geschwindigkeit

đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;đ?&#x153;&#x201D;1 = 0

Drehzahl Speed

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;1 = 0

WinkelAngular beschleunigung acceleration

1 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1 = 0 Nm

Effektivmoment RMS value torque

Effektivmoment RMS value torque Antriebsseite driving side

đ?&#x203A;źđ?&#x203A;źđ?&#x203A;źđ?&#x203A;ź1 = 0

Drehmoment Torque driving side Antriebsseite

1 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝

(0,167 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;0,085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 56 8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

= 0,058 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Tabelle 47 BewegungsgrĂśĂ&#x;en an der Antriebsseite des Getriebes fĂźr einen TĂźrantrieb Table 47 Motion variables at the driving side of the gearbox for a door motor

5 Auswahl des Motors 5.1 KenngrĂśĂ&#x;en von Elektromotoren (0,167 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;0,085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; IstEffektivmoment der Bewegungsablauf der Motorwelle bestimmt, ein konkreter Motor aus demđ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) Katalog 5 Design ofanthe motor 22 â&#x2C6;&#x2122;â&#x2C6;&#x2122;0.8đ?&#x2018; đ?&#x2018; 22 â&#x2C6;&#x2122;â&#x2C6;&#x2122;0.8đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) RMS value torque (0.167 (0.0029 (â&#x2C6;&#x2019;0.085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) 0.8đ?&#x2018; đ?&#x2018; + +wird đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )222 â&#x2C6;&#x2122;â&#x2C6;&#x2122;1.7đ?&#x2018; đ?&#x2018; 1.7đ?&#x2018; đ?&#x2018; +(â&#x2C6;&#x2019;0.085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) 0.8đ?&#x2018; đ?&#x2018; RMS value torque (0.167 (0.0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) + â&#x2C6;&#x161;unter RMS value (0.167 đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; =â&#x2C6;&#x161; = 0.058 0.058đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )2 â&#x2C6;&#x2122; 0.8đ?&#x2018; đ?&#x2018; + (0.0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; )8,3â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 1.7đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;0.085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ) ausgewĂ¤hlt. Die Auswahl erfolgt BerĂźcksichtigung der elektrischen, mechanischen und2 â&#x2C6;&#x2122; 0.8đ?&#x2018; đ?&#x2018; = Antriebsseite đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = 1đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; driving side torque driving side â&#x2C6;&#x161; 8.3đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; dargestellt werden. Die đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;1đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = Motors, die mit Hilfe von Kennlinien = 0.058 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 8.3 driving side Eigenschaften thermischen des = 0,058 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 8.3 đ?&#x2018; đ?&#x2018; KennlinienverlĂ¤ufe unterscheiden sich fĂźr dieside verschiedenen Motortypen. Die fĂźr die Auswahl des Table 4747 Motion variables at the the driving side of the the gearbox forfĂźr door drive 5.1 Characteristics of electrical motors Table 47 Motion variables at driving of gearbox for aa door drive Tabelle BewegungsgrĂśĂ&#x;en an Antriebsseite des Getriebes einen TĂźrantrieb Table relevanten 47 Motion variables atsind theder driving side of the gearbox for a door drive Motors KenngrĂśĂ&#x;en jedoch einheitlich definiert.

IfGleichstrommotor the motion sequence at the motor shaft Gleichstrommotor is determined, a specific motor is selected from BĂźrstenloser Asynchronmotor am AC-Netz Design of the the motor motor 55the Design of am DC-Netz am Pulssteller Regelung Themotor selection is made takingmit into account the electrical, mechanical and 5 catalogue. Design of the 5.1 Characteristics of electrical electrical motors motors 55.1 Auswahl des Motors 5.1 Characteristics of Characteristics of electrical motors characteristics ofmotor the motor, are represented by characteristics. the motion motion sequence atElektromotoren the shaft isiswhich determined, a specific motor motor is selected from from the the 5.1 KenngrĂśĂ&#x;en vonat IfIfthermal the sequence If the motion sequence atthe themotor motorshaft shaft isdetermined, determined,aaspecific specific motor is is selected selected from the catalogue. The selection isismade made taking into account the electrical, mechanical and thermal Ist der Bewegungsablauf an der Motorwelle bestimmt, ein konkreter Motor ausand demthermal Katalog catalogue. The selection taking account the mechanical catalogue. The selectionis made takinginto into accountwird theelectrical, electrical, mechanical and thermal characteristics ofof the motor, which are represented by characteristics. The characteristics differfor forthe the ausgewĂ¤hlt. Die Auswahl erfolgt unter BerĂźcksichtigung der elektrischen,The mechanischen und differ characteristics of the which are represented characteristics differ for the characteristics themotor, motor, which are representedby bycharacteristics. characteristics. The characteristics The characteristics differ from each motortype. However, therelevant characteristic parameters relevant different types of motors. However, the characteristic parameters relevant for the selection of the thermischen Eigenschaften des Motors, die mit Hilfe von Kennlinien dargestellt werden. Die different types of motors. However, the characteristic parameters for the selection of the different types of motors. However, the characteristic parameters relevant for the selection of the motor are uniformly defined. KennlinienverlĂ¤ufe unterscheiden sichuniformly fĂźr die verschiedenen motor are uniformly defined. for the selection of the motor are defined. Motortypen. Die fĂźr die Auswahl des motor are uniformly defined. Motors relevanten KenngrĂśĂ&#x;en sind jedoch einheitlich definiert. Gleichstrommotor DC motor on DC grid DC motor on DC grid DC motor on DC grid DC motor on DC grid am DC-Netz

Brushless DC on Brushless DC motor Brushless motor BrushlessDC DC motoron on Asynchronous motor motor on BĂźrstenloser Gleichstrommotor motoron Asynchronous on Asynchronous Asynchronmotor am AC-Netz servo amplifier with closed on servo amplifier servo amplifier with closed servo amplifier with closed am Pulssteller mit Regelung AC gridgrid on grid AC AC AC grid loop control loop with closed loopcontrol controlloop control

KenngrĂśĂ&#x;e

Beschreibung

UN :

Nennspannung

Die Spannung, die an den Motor angelegt wird. Auf diese Spannung beziehen sich alle Nenndaten in den Katalogen. Die Motoranwendung ist jedoch nicht auf diese Spannung beschrĂ¤nkt.

IN:

Nennstrom

Der Strom, der durch den Motor flieĂ&#x;t, wenn er bei Nenndrehmoment und Nenndrehzahl betrieben wird.

MN:

Nenndrehmoment

Das Drehmoment, das der Motor bei Nennumgebungstemperatur und bei Betrieb mit Nenndrehzahl dauerhaft ohne Ă&#x153;berhitzung abgeben kann.

Characteristic Characteristic Characteristic Anlaufmoment M A: KenngrĂśĂ&#x;e Rated voltage Rated voltage UUNNU::N : Rated voltage Mmax: Maximales Nennspannung UN : Drehmoment Rated current Rated current IINN:I: N: Rated current

Das Drehmoment, das der Motor im Stillstand bei Drehzahl 0 maximal Description Description Description abgeben kann. Das Anlaufmoment von seinem theoretischen Wert Thevoltage voltage applied tothe themotor. motor.kann All rated rated values in the The voltage applied to rated values inbegrenzt the Beschreibung The applied All values the abweichen, wenn die to integrierte Elektronik den Stromin oder der catalogues refer to this voltage. However, the motor is not limited catalogues refer to to this this voltage. However, However, the the motor motor isis not not limited limited catalogues refer voltage. Motor entmagnetisiert wurde. tothis this voltage.die an den Motor angelegt wird. Die Spannung, to this voltage. to voltage. Das Drehmoment, mit dem der Motor maximal betrieben werden kann Auf Spannung beziehen sich alle Nenndaten inififden Katalogen. Thediese absorption current flowing through the motor at The absorption current flowing through theder motor operated at unter BerĂźcksichtigung des Pulsstellers und Entmagnetisierung. The absorption current motor if operated operated at Die Motoranwendung istflowing jedoch through nicht aufthe diese Spannung beschrĂ¤nkt. rated torque and rated speed.

Design of the motor Kenngröße Characteristic

Beschreibung Description

UN :

Nennspannung Rated voltage

Die Spannung, die an den Motor angelegt wird. The voltage applied to the motor. All rated values in the catalogues refer Auf diese Spannung beziehen sich alle Nenndaten in den Katalogen. to voltage. However, motor is not to this voltage. Diethis Motoranwendung ist the jedoch nicht auf limited diese Spannung beschränkt.

IN:

Nennstrom Rated current

The flowing through Der absorption Strom, der current durch den Motor fließt, the motor if operated at rated wenn er beirated Nenndrehmoment und Nenndrehzahl betrieben wird. torque and speed.

MN:

Nenndrehmoment Rated torque

The the motor permanently at rated ambient tempeDastorque, Drehmoment, dascan derprovide Motor bei Nennumgebungstemperatur und rature duringmit operation at rated speed without overheating.abgeben kann. bei Betrieb Nenndrehzahl dauerhaft ohne Überhitzung

MA:

Anlaufmoment Start-up torque

Dasmaximum Drehmoment, dasthe dermotor Motorprovides im Stillstand bei Drehzahl 0 maximal The torque, at standstill. The start-up abgeben kann. Das Anlaufmoment kann von seinem theoretischen Wert torque can deviate from its theoretical value if the control unit abweichen, wenn die integrierte Elektronik den Strom begrenztlimits oderthe der current or the motor has been demagnetized. Motor entmagnetisiert wurde.

Mmax:

Maximales Maximum torque Drehmoment

MK :

Kippmoment Stall torque

nN:

Nenndrehzahl Rated speed

n0:

Leerlaufdrehzahl No-load speed

nmax:

Maximalespeed Drehzahl Maximum

PN:

Nennleistung Rated power

Pmax:

Maximum Maximalepower Leistung

kT:

Torque constant

M R:

Reibmoment Friction torque

Drehmomentkonstante

The at whichmit thedem motor be operated its maximum taking Dastorque Drehmoment, dercan Motor maximal at betrieben werden kann unter Berücksichtigung des Pulsstellers und der Entmagnetisierung. into account the pulse controller and demagnetization The torque anein asynchronous motorbei can provide if it is operaDasmaximum Drehmoment, das Asynchronmotor Betrieb am on Wechseloder maximal abgeben kann. ted a 1-phase orDrehstromnetz 3-phase AC grid. The of the which results operation at rated voltage Die speed Drehzahl desmotor Motors, die sich beiduring Betrieb mit Nennspannung undrated bei Abgabe and torque.des Nenndrehmoments einstellt. The of the which results during operation without load Die speed Drehzahl, diemotor sich einstellt, wenn der Motor ohne Last mit Nennat rated voltage. The wird. no-load may deviatekann fromvon its ihrem theoretical spannung betrieben Diespeed Leerlaufdrehzahl theoretischen abweichen, wenn in Lagernshaft der Motorwelle value if friction Wert occurs in the bearings of den the motor or due to a oder fan aufgrundto eines an die shaft. Motorwelle angebauten Lüfters Reibung auftritt. attached the motor The at which the motor can be operated at itswerden maximum without Die speed Drehzahl, mit der der Motor maximal betrieben kann, ohne dass mechanische Beschädigungen auftreten. mechanical damage. The output power of the motor can die provide permanently Die mechanical mechanische Abgabeleistung des Motors, er bei Nennat rated ambient temperature without overheating. It is calculated from umgebungstemperatur dauerhaft ohne Überhitzung abgeben kann. Sie berechnet sich austorque. Nenndrehzahl und Nenndrehmoment. rated speed and rated The output power, the motor can provide for a Die maximum maximale mechanical mechanische Abgabeleistung, die der Motor kurzzeitig 57 abgeben short time.kann. The between the motor current and motor torque. Dasratio Verhältnis zwischen Motorstrom undthe dem Motordrehmoment. Die torque Drehmomentkonstante bei Gleichstrommotoren und DC The constant is used wird in brushed DC motors, brushless Synchronmotoren bzw. bürstenlosen Gleichstrommotoren verwendet. motors and synchronous motors. Torque, which must be overcomewerden if the current-less motor is drivenMotor from Drehmoment, das überwunden muss, falls der stromlose von außen angetrieben wird. outside.

Tabelle 48 Kenngrößen für Elektromotoren Table 48 Characteristics for electrical motors

Die Auswahl des Motors erfolgt in Abhängigkeit von der Anwendung auf zwei verschiedenen Wegen:

Auswahl nach Drehzahlund Leistungsbedarf • Konstantantriebe: Depending on the application the selection of the motor takes place in two different ways: •

Drehzahlveränderliche Antriebe und

Auswahl nach Drehzahl- und

mit Motornachrechnung und Servoantriebe: • Constant speed motors: Selection accordingDrehmomentbedarf to speed and power requirements thermischer Auslegung

• Variable speed motors and servo motors: Selection according to speed and torque requirements with motor recalculation and thermal verification

5.2 Konstantantriebe Konstantantriebe werden direkt am Gleichspannungs- oder Wechselspannungsnetz betrieben und sind in ihrerConstant Drehzahl nicht veränderlich. 5.2 speed motorsSieht man vom Anlaufvorgang ab, sind keine Drehzahländerungen und damit keine Beschleunigungs- und Bremsphasen zu berücksichtigen. Die Auswahl des Motors erfolgt deshalb am einfachsten über den stationären Arbeitspunkt, an dem der Motor betrieben wird. Constant speed motors are operated directly on the DC or AC voltage grid and run at fixed

speed. Neglecting the start-up, no speed changes and thus no acceleration and deceleration 5.2.1 Konstantantriebe permanent Gleichstrommotor phases have to be takenmit into account. erregtem The selection of the motor can be done most easily by Wird ein permanent erregter Gleichstrommotor an einer konstanten Versorgungsspannung betrieben

consideration theLast operating point the motor isdurch running at. und ändert sich of seine nicht, kann sein Verhalten die nachfolgenden Gleichungen

beschrieben werden. Die Auswahl des Motors erfolgt an Hand des Arbeitspunktes, an dem der Konstantantrieb betrieben wird.

Kennlinien des Gleichstrommotors

thermal verification 5.2 Konstantantriebe Konstantantriebe werden direkt am Gleichspannungs- oder Wechselspannungsnetz betrieben und sind in ihrer Drehzahl nicht verĂ¤nderlich. Sieht man vom Anlaufvorgang ab, sind keine Design of the motor 5.2 Constant speedund drives DrehzahlĂ¤nderungen damit keine Beschleunigungs- und Bremsphasen zu berĂźcksichtigen. Die Constant driveserfolgt are operated on the DC or den AC voltage grid Arbeitspunkt, speed. and run at fixed Auswahl speed des Motors deshalb directly am einfachsten Ăźber stationĂ¤ren an dem der Neglecting the start-up, Motor betrieben wird. no speed changes and thus no acceleration and deceleration phases are to 5.2.1 Constant speed motors with permanent magnet DC motor be taken into account. The selection of the motor can be done most easily by consideration of the operating point the motor is running at. 5.2.1 Konstantantriebe mit permanent erregtem Gleichstrommotor If a permanent magnet DC motor is operated at a constant supply voltage under steady state Wird ein permanent erregter Gleichstrommotor an einer konstanten Versorgungsspannung betrieben und Ă¤ndert sich Last nicht, kann sein Verhalten durch die motor nachfolgenden 5.2.1 Constant speed drives permanent magnet DC conditions, its seine behaviour canwith be described by the following equations.Gleichungen The DC motor is Motorsaterfolgt an Hand des voltage Arbeitspunktes, an dem der Ifbeschrieben a permanentwerden. magnetDie DCAuswahl motor isdes operated a constant supply under steady state selected basing on the operating point at which the constant motor is running. Konstantantrieb betrieben wird. conditions, its behaviour can be described by the following equations. The DC motor is selected based on the operating point at which the constant drive is running. Kennlinien des Gleichstrommotors Characteristics of the DC motor Characteristics of the DC motor

â&#x2030;¤đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ â&#x2030;¤ đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ â&#x2030;¤â&#x2030;¤đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Auswahlbedingungen Selection conditions conditions Selection

đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;

Motordrehmoment MM: Motor torque MM: Motor torque

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ??źđ??źđ??źđ??ź â&#x2C6;&#x2122; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;= đ??źđ??źđ??´đ??´đ??´đ??´đ??´đ??´â&#x2C6;&#x2122; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

Ankerspannung(Versorgungsspannung) UA: Armature voltage (supply voltage) UA: Armature voltage (supply voltage)

đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; = đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; â&#x2C6;&#x2122; đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ??´đ??´đ??´đ??´đ??´đ??´= đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ??´đ??´đ??´đ??´đ??´đ??´â&#x2C6;&#x2122; đ??źđ??źđ??´đ??´đ??´đ??´đ??´đ??´+ đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;

Motorspannung (EMK) UM: Motor voltage (EMF) UM: Motor voltage (EMF) current IAnlauf: Start-up Anlaufstrom fĂźr nat=n=0 0 IStart-up: Start-up current at n=0

Elektrische Leistung P1: Electrical power P1: Electrical power

Mechanische Leistung an der Motorwelle Mechanical power at motor shaft P2: P2: Mechanical power at motor shaft

Efficiency Wirkungsgrad

đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; â&#x2C6;&#x2122; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; â&#x2C6;&#x2122; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ??´đ??´đ??´đ??´ đ??źđ??źđ??źđ??źđ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ??´đ??´ đ??źđ??źđ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;â&#x2C6;&#x2019;đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘ = đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ??´đ??´đ??´đ??´ đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ??´đ??´ đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;1 = đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ??´đ??´đ??´đ??´ â&#x2C6;&#x2122; đ??źđ??źđ??źđ??źđ??´đ??´đ??´đ??´ đ?&#x2018;&#x192;đ?&#x2018;&#x192;1 = đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ??´đ??´ â&#x2C6;&#x2122; đ??źđ??źđ??´đ??´

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Verlustleistung im in elektrischen Kreis PVerl: Power dissipation the armature winding

Îˇ:

đ?&#x2018; đ?&#x2018;

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Motordrehzahl Motor speed nMn: M: Motor speed

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đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;2 đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;1

Leerlaufdrehzahl Nenndrehmoment n0: MN: Rated No-load speed (1/s) torque Nenndrehzahl Motordrehmoment am Arbeitspunkt speed (1/s) nN: Rated MAP: Motor torque at operating point Motordrehzahl Arbeitspunkt Ankerstrom speed at am operating point (1/s) current nAP: Motor IA: Armature Drehmomentkonstante Ankerwiderstand Torque constant resistance kT: RA: Armature Tabelle Gleichungen des Gleichstrommotors im stationĂ¤ren Betrieb Table 49 49 Equations of the DC motor in steady-state operation Zu beachten ist, dass die Versorgungspannung in einem bestimmten zulĂ¤ssigen Bereich gewĂ¤hlt werden kann. Ist die Wicklungsisolation der Motoren entsprechend ausgelegt, kĂśnnen diese Motoren beispielsweise an Versorgungsspannungen von 12 V oder 24 V betrieben werden. FĂźr jede dieser Versorgungsspannungen gilt ein gesondertes Mitoperated, steigender for winding insulation of the motors is designed accordingly,Kennliniendiagramm. these motors can be Versorgungsspannung wird die Drehzahl-Drehmoment-Kennlinie des permanent erregten Gleichstrommotors nach oben verschoben. Die Leerlaufdrehzahl und die Leistung des Motors steigen nĂ¤herungsweise proportional zur Versorgungsspannung an.

It should be noted that the supply voltage can be selected within a certain permissible range. If the

Wird der stillstehende permanent erregte Gleichstrommotor an die DC-Spannung zugeschalten, flieĂ&#x;t

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Wirkungsgrad Îˇ: ď ¨: Efficiency Design of the motor Leerlaufdrehzahl n0:

MN: nn0N:: No-load speed MAPN:: Motordrehmoment Rated torque Nenndrehzahl am Arbeitspunkt M nnNAP: : Rated speed Motor torque at operating point Motordrehzahl am Arbeitspunkt IM A: OP: Ankerstrom nkOP Motor speed atvoltages operatingof point Armature currentdiagram applies to each Drehmomentkonstante Ankerwiderstand RIAA: example, at supply 12 V or 24 V. A separate characteristic T: : kT: Torque constant des Gleichstrommotors imRstationĂ¤ren Armature resistance A: Tabelle 49 Gleichungen Betrieb ofTable these supply voltages. As the supply voltage increases, the speed-torque characteristic of 49 Equations of the DC motor in steady-state operation Zu beachten ist, dass die Versorgungspannung in einem bestimmten zulĂ¤ssigen Bereich gewĂ¤hlt the permanent magnet DC motor is shifted upwards. The no-load speed and the power of the It should be noted the supply voltage be selected within aausgelegt, certain permissible range. If the werden kann. Ist diethat Wicklungsisolation der can Motoren entsprechend kĂśnnen diese Motoren motor increase approximately proportionally to the supply voltage. winding insulation of the motors is designedvon accordingly, motors can be operated, for example, beispielsweise an Versorgungsspannungen 12 V oder these 24 V betrieben werden. at supply voltages of 12 V or 24 V. FĂźr jede dieser Versorgungsspannungen gilt ein gesondertes Kennliniendiagramm. Mit steigender A separate characteristic applies to each of these supply As erregten the supply voltage Versorgungsspannung wirddiagram die Drehzahl-Drehmoment-Kennlinie desvoltages. permanent If a permanent magnet DC motor in standstill is connected to die the DC voltage, high start-up curGleichstrommotors nach obencharacteristic verschoben. Die Leerlaufdrehzahl und Motors steigenThe increases, the speed-torque of the permanent magnet DCLeistung motor isdes shifted upwards. nĂ¤herungsweise proportional zur Versorgungsspannung an. no-load speed and the power of the motor increase approximately proportionally to the supply voltage. rent flows. This can be limited by additional series resistors, which are bypassed successively.

By the resistance stages, the peak values ofan and the start-up Wird der stillstehende permanent erregte Gleichstrommotor die DC-Spannung zugeschalten, flieĂ&#x;t can If aselecting permanent magnet DC motor in standstill is connected tothe thetorque DC voltage, high start-up current current ein hoher Anlaufstrom. Dieser kann durchseries AnkervorwiderstĂ¤nde begrenzt werden, die nacheinander flows. This can limited by additional resistors, which are bypassed successively. By selecting be adjusted as be required ĂźberbrĂźckt werden. Durch Widerstandstufen kannthe diestart-up Schwankungsbreite des the resistance stages, theWahl peakder values of the torque and current can be adjusted as Drehmomentes und des Anlaufstromes je nach Bedarf eingestellt werden. required. Anlaufschaltung mit AnkervorwiderstĂ¤nden Start-up circuit with series resistors Start-up circuit with series resistors

ď ŹÎť::

Switching ratio SchaltverhĂ¤ltnis Switching ratio

z: z:

Number of switching stages Anzahl der Schaltstufen Number of switching stages

Kennlinien des Gleichstrommotors Characteristics of the DC motor

Characteristics of the DC motor

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Îťď Ź==

Vorschaltwiderstand 1 resistor 11 V1:: Series RRV1 Series resistor

Vorschaltwiderstand 2 resistor 22 V2:: Series RRV2 Series resistor

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Vorschaltwiderstand n resistor nn Vn:: Series RRVn Series resistor Sum of series resistors Summe der VorschaltwiderstĂ¤nde RRVVď &#x201C;ÎŁ:: Sum of series resistors

PVerl: Power dissipation in the series resistors PLoss : Power dissipation in the series resistors

. nn00::

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I A: Armature current I A: Armature current Imax: Maximum armature current at start-up Imax: Maximum armature current at start-up Imin: Minimum armature current at start-up I : Armature Minimumresistance armature current at start-up Rmin A: Minimum motor torque at start-up RA: Torque Armature resistance constant kT: kT: Torque constant Tabelle 50 Gleichungen des Gleichstrommotors im stationĂ¤ren Betrieb mit 57 AnkervorwiderstĂ¤nden Table 50 Equations of the DC motor in steady-state operation with series resistors nN: nN: Mmax: Mmax: Mmin: Mmin:

No-load speed No-load speed Rated speed Rated speed Maximum motor torque at start-up Maximum motor torque at start-up Minimum motor torque at start-up

5.2.2 Konstantantriebe mit Asynchronmotor Wird ein Asynchronmotor an einem Wechsel- oder Drehstromnetz betrieben und Ă¤ndert sich seine

Mmax: Mmin:

Imin: RA : kT: Tabelle 50 Gleichungen des Gleichstrommotors im stationĂ¤ren Betrieb mit AnkervorwiderstĂ¤nden

5.2.2 5.2.2

Design of the motor

Constant speed motors with asynchronous motor Konstantantriebe mit Asynchronmotor

IfWird an asynchronous motoraniseinem operated on a 1-phase or 3-phasebetrieben AC grid under steady-state ein Asynchronmotor Wechseloder Drehstromnetz und Ă¤ndert sich seineconditions Last nicht, kann sein durch die following nachfolgenden Gleichungen beschrieben werden. Die its behaviour can beVerhalten described by the equations:

Auswahl des Motors erfolgt an Hand des Arbeitspunktes, an dem der Konstantantrieb betrieben wird.

Characteristics of the asynchronous motor Kennlinien des Asynchronmotors

Operation point

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Selection conditions Auswahlbedingungen

Slip

n0:

No-load speed

MM:

Motor torque

Pel:

đ?&#x2018; đ?&#x2018; =

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Electrical power (active power)

đ?&#x2018; đ?&#x2018;

đ?&#x2018; đ?&#x2018; đ??žđ??ž

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Pmech: Mechanical power at motor shaft

ď ¨:

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Efficiency

p: ď Ş: Polpair number of the motor Power factor fN I S: Frequency of the grid Stator current UN: Voltage amplitude of the grid I0: No-load current n0: No-load speed (1/s) MN: Rated torque nN: Rated speed (1/s) MA: Start-up torque nAP: Motor speed at operating point MK: Stall torque Stall slip sK : MAP: Motor torque at operating point Tabelle 51 Gleichungen des Asynchronmotors im stationĂ¤ren Betrieb Table 51 Equations of the asynchronous motor in steady-state operation 60

The asynchronous motor is selected basing on the operating point at which the constant motor is running. If a 3-phase asynchronous motor in standstill is connected to the AC grid, high start-up current flows. This can be reduced to 1/3 by a star-delta start-up circuit. The star-delta

Design of the motor Wird der stillstehende Drehstromasynchronmotor an das AC-Netz zugeschalten, fließt ein hoher

start-up circuit cankann be used asynchronousauf motors whose rated voltage in delta Anlaufstrom. Dieser durch for eine3-phase Stern-Dreieckschaltung 1/3 abgesenkt werden. Die Stern-

Dreieckschaltung ist fürtoDrehstromasynchronmotoren connection is equal the grid voltage at which einsetzbar, the motor deren has toNennspannung be operated in and which must Dreieckschaltung gleich ist der Netzspannung, an der der Motor betrieben werden soll und die nur

only against a low load torque. gegenstart-up ein geringes Lastmoment anlaufen müssen. Stern-Dreieckanlauf Star-delta start-up

Kennlinien des Asynchronmotors Characteristics of the asynchronous motor

Schaltung des Motors im Anlauf Wiring of the motor during start-up

Sternschaltung Star connection

Schaltung des Motors im Betrieb Wiring of the motor during operation

Dreieckschaltung Delta connection 1 3

Reduktionstart-up Anlaufstrom Lowering current in in Sternschaltung star connection to auf

1 3

Lowering motor torque in Reduktionstart-up Motordrehmoment star connection to auf in Sternschaltung

Leerlaufdrehzahl Statorstrom n0: No-load IS: Stator current speed Nenndrehzahl Leerlaufstrom current nN: Rated I0: No-load speed current Nenndrehmoment Anlaufstrom torque MN: Rated IA: Start-up torque Anlaufmoment MA: Start-up Tabelle 52 Gleichungen des Asynchronmotors im stationären Betrieb mit SternTable 52 Equations of the asynchronous motor in steady-state operation with star-delta start-up circuit Dreieckschaltung

5.3 Variable speed motors and Servo motors 5.3 Drehzahlveränderliche Servoantriebe 5.3.1 Selection of theAntriebe motor und according to mechanical parameters 5.3.1 Auswahl nach mechanischen Kenngrößen Die Auswahl nach mechanischen Kenngrößen erfolgt in zwei Schritten

The selection according to mechanical parameters is carried out in two steps: 1. Vorauswahl des Motors

1. Preselection of the motor 2. Motornachrechnung unter Berücksichtigung des Trägheitsmomentes des Motors 2. Recalculation considering the moment of inertia of the motor

Sind die Bewegungsabläufe an der Antriebsseite des Getriebes und damit an der Motorwelle entsprechend Kapitel 4 ermittelt, wird im ersten Schritt der konkrete Motor aus dem Katalog ausgewählt. Dazu werden die markanten des Lastspiels in das If the motion sequence at the driving Arbeitspunkte side of the gearbox and thus atKennliniendiagramm the motor shaft are des Motors übertragen. Dabei werden die Vorzeichen des Drehmoments und der Drehzahl determined in accordance with chapter 4, the motor is selected from the catalogue in the first vernachlässigt.

step. For this purpose, the relevant operating points of the load cycle are transferred into the characteristic diagram of the motor. The signs of the torque and the speed are neglected.

Design of the motor Lastspiel Load cycle

Kennlinien des Motors Characteristics of the motor

Lastspiel

Kennlinien des Motors

Alle markanten Arbeitspunkte innerhalb Allrelevant relevant operating points are areliegen located within the the All operating points located within areaKennlinie limited by byfĂźr theden characteristic for cyclic operation. der zyklischen Betrieb. Selection condition area limited the characteristic for cyclic All operating points areliegen located within the Allerelevant markanten Arbeitspunkte innerhalb Auswahlbedingung Nenndrehmoment des Motors Rated torque of the motor operation. M N: limited condition area by theden characteristic cyclic der Kennlinie fĂźr zyklischen for Betrieb. MSelection Motordrehmoment fĂźr die Vorauswahl torque for preselection of the motor MV: Motor torque of the motor Maximales Drehmoment des Motors MNmax MMV: Motor torque for preselection of the motor M : : Maximum Rated torque of the motor operation. des Motors Nenndrehmoment des Motors MN: Rated speed of the motor Nenndrehzahl :: : Maximum nNmax MM Motordrehmoment fĂźr die Vorauswahl MV :: Motor speed M torque of Motors the motor Motor torque for preselection of the motor M Rated torque ofdes the motor speed M:MV: Motor Motordrehzahl nnM Maximales Drehmoment des Motors : Maximale MNmax speed of motor the motor Drehzahl des Motors des Motors nNmax nM : : : Maximum Rated speed of the nM: Motor speed Maximum torque the motor Nenndrehzahl desofMotors : nNmax Motordrehzahl nM: Tabelle 53 Vorauswahl des Motors nach mechanischen KenngrĂśĂ&#x;en nnmax : Maximum speed of the motor : : Rated speed of the motor Maximale Drehzahl des Motors nNmax Table 53 53 Preselection of theof motor motor according to mechanical parameters Table Preselection according to mechanical parameters nmax:KenngrĂśĂ&#x;en Maximum speed of the motor Tabelle 53 Vorauswahl desthe Motors nach mechanischen Liegen alle markanten Arbeitspunkte des Lastspiels innerhalb der Kennlinie fĂźr den Table 53 Preselection of the motor according to mechanical parameters zyklischen Betrieb, ist der Motor unterpoints mechanischen Gesichtspunkten geeignet. If all relevant operating of thedes load cycle areinnerhalb located within the areafĂźr limited by the characteristic Liegen alle markanten Arbeitspunkte Lastspiels derwithin Kennlinie den zyklischen If all relevant operating points ofsuitable the load cycle are located the area limited by the characteriDiese Motoren werden vorausgewĂ¤hlt in den Auslegungsschritten weiter for cyclic operation, the motor is from afolgenden mechanical point of If all relevant operating points of theund loadGesichtspunkten cycle are located within theview. area limited by betrachtet. the characteristic Betrieb, ist der Motor unter mechanischen geeignet. Alle anderen Motoren werden aussortiert. These motors are preselected and further considered in the following selection steps. All other motors stic for cyclic operation, the motor is suitable from a mechanical point of view. These motors are for cyclic operation, motor is suitable mechanical point of view. Diese Motoren werdenthe vorausgewĂ¤hlt und infrom den afolgenden Auslegungsschritten weiter betrachtet. are neglected. These motors are preselected and further considered in the following selection steps. All other motors Alle anderen Motoren werden aussortiert. preselected and further considered in the selectiondes steps. AllberĂźcksichtigt. other motors are neglected. In are der neglected. Motornachrechnung wird zusĂ¤tzlich dasfollowing TrĂ¤gheitsmoment Motors theMotornachrechnung motorrecalculation, recalculation,wird the moment of of thethe motor isdes additionally takentaken into account. In the motor thezusĂ¤tzlich momentofdas ofinertia inertia motor isMotors additionally into account. InIn der TrĂ¤gheitsmoment berĂźcksichtigt. In the motor recalculation, the moment of inertia of the motor is additionally Lastspiel Kennlinien des Motorstaken into account. Load cycle Characteristics of the motor Lastspiel Kennlinien des Motors Load Characteristics themotor motor Loadcycle cycle Characteristics ofofthe Auswahlbedingung Selection condition

M Motordrehmoment fĂźr die MMN Motor torque for recalculation of the motor MN:: torque for recalculation Motornachrechnung MMN: Motor Motordrehmoment fĂźr die of the motor MMN: Motor torque for recalculation of the motor Motornachrechnung Auswahlbedingung Selection condition Auswahlbedingung Selection condition Selection condition M : Motordrehmoment fĂźr die Vorauswahl des MMV Motor torque for preselection of the motor MV: MMV: Motor Motordrehmoment fĂźr die Vorauswahl des torque for preselection Motors nMMMV : : Motor speed Motor torque for preselection of the motor Motors of Motordrehzahl the motor nM M: Rated torque motor N: Motor speed of thedes speed Motordrehzahl nnMNM::: Motor Nenndrehmoment Motors M M : Rated Maximum torque of the motor max M : Rated torque of the motor N torque of the motor Nenndrehmoment des Motors MN: Tabelle 54 Motornachrechnung Table 54 Motor recalculation M : Maximum torque of the max 54 Motornachrechnungmotor Tabelle Table 54 Motor recalculation Table 54 Motor recalculation

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In each segment of the load cycle where acceleration or deceleration occurs, the torque applied by the 62 62 motor changes. result, a corrected load cycle whose relevant operating points are again In each segmentAsofathe loadthere cycleiswhere acceleration or deceleration occurs, the torque applied by the transferred to theAs characteristic diagram of the motor. motor changes. a result, there is a corrected load cycle whose relevant operating points are again transferred to the characteristic diagram of the motor. If all relevant operating points of the load cycle are still located within the area limited by the

Design of the motor In each segment of the load cycle where acceleration or deceleration occurs, the torque applied by the motor changes. As a result, there is a corrected load cycle whose relevant operating points are again transferred to the characteristic diagram of the motor. If all relevant operating points of the load cycle are still located within the area limited by the characteristic for cyclic operation the motor is finally suitable from a mechanical point of view. This motor is selected and further considered in the following selection steps. If some operating points are now outside the area limited by the characteristic for cyclic operation, the preselected motor is not suitable. These motors are neglected and no longer included in the following selection steps. The mechanical parameters is thus completed. FĂźr jeden Abschnitt desselection Lastspiels,according in dem einetoDrehzahlĂ¤nderung durch Beschleunigen oder

Bremsen auftritt, Ă¤ndert sich das vom Motor aufzubringende Drehmoment. Als Ergebnis liegt ein korrigiertes Drehmoment-Zeit-Diagramm vor, dessen markante Punkte wieder in das 5.3.2 Selection the motor according Kennliniendiagramm desof Motors Ăźbertragen werden. to thermal characteristics Liegen alle Punkte des Lastspiels immer noch innerhalb der Kurve fĂźr den zyklischen Betrieb, sind die vorausgewĂ¤hlten Motoren unter mechanischen Gesichtspunkten in der Lage, das geforderte Lastspiel The energyLiegen conversion within jetzt the motor is subject tofĂźr loss, leads to temperature rise zu erfĂźllen. einige Punkte auĂ&#x;erhalb der Kurve denwhich zyklischen Betrieb, sind die betreffenden vorausgewĂ¤hlten Motoren nicht geeignet. Diese Motoren werden aussortiert und in die within the motor. It has to be ensured that the heating does not exceed the permissible limits folgenden Auslegungsschritte nicht mehr mit einbezogen. according to Table 55. Die Auslegung nach mechanischen KenngrĂśĂ&#x;en ist damit abgeschlossen. 5.3.2 Auswahl thermischen KenngrĂśĂ&#x;en Thermal classnach according to DIN-VDE 0530 A E MotorsBfĂźhrt. Es muss F Die Energiewandlung im Motor ist verlustbehaftet, was zur ErwĂ¤rmung des (IEC 60085) sichergestellt werden, dass die ErwĂ¤rmung die zulĂ¤ssigen Grenzwerte nach Tabelle 55 nicht Ăźberschreitet. Maximum permissible motor temperature

at the hottest point of the winding in Â°C

105

120

100

115

WĂ¤rmeklasse nach DIN-VDE 0530 (IEC 60085)

Average permissible motor temperature in Â°C

HĂśchste zulĂ¤ssige Motortemperatur am heiĂ&#x;esten Punkt der Wicklung in Â°C

Permissible over temperature at an ambient Mittlere zulĂ¤ssige Motortemperatur in Â°C temperature of 40 Â°C

ZulĂ¤ssige Ă&#x153;bertemperatur bei einer Umgebungstemperatur Table 55 Limits of motor heating von 40 Â°C

105

130

155

120 120

H 180

140 130

100

115

120

165

155

180

100

125

140

165

100

125

Tabelle 55 Grenzwerte der MotorerwĂ¤rmung Bei Belastung des Motors steigtitsseine Temperatur zeitverzĂśgert Die zeitliche VerzĂśgerung wirdis deterWhen the motor is loaded, temperature increases withan. a time delay. The time delay durch die thermische Zeitkonstante des Motors bestimmt.

mined by the thermal time constant of the motor.

Motor Motortemperatur temperature

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Nacht=3â&#x2039;&#x2026;T t = 3thâ&#x2039;&#x2026; T, th hat der Motor 9595 %% seiner After the motor reached of its stationĂ¤ren Ă&#x153;bertemperatur steady-state over temperature.erreicht. Umgebungstemperatur temperature Ď&#x2026;0: Ambient Ď&#x2026;Ov: steady-state temperature Endtemperatur Tth: Ď&#x2026;st: FinalStationĂ¤re Tabelle 56 Temperaturverlauf des Motors unter Last Table 56 Temperature response of the loaded motor

Ă&#x153;bertemperatur Over temperature Thermal time constant of thedes motor Thermische Zeitkonstante Motors

Die Dauerlastkennlinie des Motors gibt das thermisch zulĂ¤ssige Dauerdrehmoment als Funktion der Drehzahl an. Da drehzahlverĂ¤nderliche Antriebe und Servoantriebe eine zeitlich verĂ¤nderliche Belastung aufweisen, muss diese auf eine Ersatzdauerlast bzw. einen Ersatzarbeitspunkt umgerechnet werden. Die Ersatzdauerlast erwĂ¤rmt den Motor in gleicher Weise wie das reale Lastspiel.

ďż˝

Nach t = 3â&#x2039;&#x2026; Tth hat der Motor 95 % seiner stationĂ¤ren Ă&#x153;bertemperatur erreicht.

Design of the motor Umgebungstemperatur Ă&#x153;bertemperatur Ď&#x2026;0 : Ď&#x2026;Ov: Thermische Zeitkonstante des Motors StationĂ¤re Endtemperatur Tth: Ď&#x2026;st: Tabelle 56 Temperaturverlauf des Motors The permanent load characteristic curve unter of theLast motor indicates the thermally permissible continuous torque as a function of the speed. Since variable speed motors and servo motors have a Die Dauerlastkennlinie des Motors gibt das thermisch zulĂ¤ssige Dauerdrehmoment als Funktion der

Drehzahl an.variable Da drehzahlverĂ¤nderliche Antriebe und Servoantriebe eineequivalent zeitlich verĂ¤nderliche temporally load, this must be converted to a continuous load or an equivalent Belastung aufweisen, muss diese auf eine Ersatzdauerlast bzw. einen Ersatzarbeitspunkt

operating point. The Die continuous equivalent torque thegleicher motor Weise in the wie same umgerechnet werden. Ersatzdauerlast erwĂ¤rmt den heats Motor in dasway realeas the torque Lastspiel. of the real load cycle. The continuous equivalent torque and the equivalent operating point are Die Ersatzdauerlast bzw. der Ersatzarbeitspunkt werden durch das Effektivmoment und die mittlere

described by the beschrieben. RMS value of the torque and the average absolute speed. absolute Drehzahl Lastspiel Load cycle

Kennlinien des Motors Characteristics of the motor

Continuous equivalent loadErwĂ¤rmung Ersatzdauerlast mit gleicher causing same heating

Meff:

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RMS value of the torque, general Effektivmoment, allgemeine Definition definition RMS value of the torque, simplified Effektivmoment, vereinfachte Formel formula for segment wise constant fĂźr abschnittsweise konstantes motor torque Motordrehmoment

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Mittlere absolute Drehzahl, allgemeine Definition

Mean absolute speed, general definition

Auswahlbedingung Selection condition Randbedingung: Boundary condition:

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Der equivalent Ersatzarbeitspunkt Kennlinie The operating liegt pointunterhalb is locatedder within the area limited the characteristic for permanent operation fĂźr denby Dauerbetrieb.

Nenndrehzahl nN: Rated speed ofdes the Motors motor (1/s) Maximale speed Drehzahl desmotor Motors nmax: Maximum of the (1/s) torque within im segment I, II, I, III,II,IVIII, IV Motordrehmoment Abschnitt MI, II, III, IV: Motor of Abschnitts segment I, II, III,III, IV IV Dauer des I, II, TI, II, III, IV: Duration Duration of Lastspiels load cycle T: Dauer des Thermal timeZeitkonstante constant of the motor Thermische des Motors Tth: Tabelle 57 Auswahl des Motors nach thermischen KenngrĂśĂ&#x;en MMN:

Motordrehmoment aus der Motor torque determined during motor Motornachrechnung recalculation Motordrehzahl nM: Motor speed (1/s) Nenndrehmoment des Motors MN: Rated torque of the motor Maximales Drehmoment des Motors torque of the motor Mmax: Maximum

Table 57 Selection of the motor according to thermal characteristics

Lieg der Ersatzarbeitspunkt unterhalb der Kennlinie fĂźr den Dauerbetrieb, ist der entsprechende Motor unter thermischen Gesichtspunkten geeignet.

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Mittlere absolute Drehzahl, Design ofallgemeine the motorDefinition

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

Auswahlbedingung Der Ersatzarbeitspunkt liegt unterhalb der Kennlinie fĂźr den đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; â&#x2030;¤ 0,1 â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąâ&#x201E;&#x17D;of the load cycle IfRandbedingung: the equivalent operating point is Dauerbetrieb. located within the area limited by the chaNenndrehzahl des Motors Motordrehmoment aus der racteristic for permanent operation the motornNis: suitable from a thermal point of view. MMN: Maximale Drehzahl des Motors nmax: Motornachrechnung MI, II, III, IV: Motordrehmoment im Abschnitt I, II, III, IV Motordrehzahl nM: TI, II, III, IV: Dauer des Abschnitts I, II, III, IV Nenndrehmoment Motors N: boundary IfMthe conditiondes Tâ&#x2030;¤ 0,1â&#x2C6;&#x2122;T all relevant operating points of the load cycle th is not fulfilled, T: Dauer des Lastspiels Mmax: Maximales Drehmoment des Motors Thermische Zeitkonstante Motors This have to be located within the area limited by Tthe for permanent des operation. th: characteristic Tabelle 57 Auswahl des Motors nach thermischen KenngrĂśĂ&#x;en

approach guarantees a safe motor selection, but may lead to an over sizing in some cases. In Lieg der Ersatzarbeitspunkt unterhalb derthermal Kennlinie fĂźr den Dauerbetrieb, ist is derrecommended entsprechende Motor uncertain cases, a simulation of the behaviour of the motor or carry unter thermischen Gesichtspunkten geeignet.

out tests in the actual application.

Sollte die Randbedingung đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018;&#x2021; 0,1 â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ąđ?&#x2018;Ą nicht erfĂźllt sein, mĂźssen alle markanten Arbeitspunkte des Lastspiels unter der Kennlinie fĂźr den Dauerbetrieb liegen. Dieser Ansatz garantiert eine sichere 5.3.3 the encoder Auslegung,Selection kann aber imof Einzelfall auch zu einer Ă&#x153;berdimensionierung fĂźhren. In GrenzfĂ¤llen ist eine Simulation des thermischen Verhaltens des Motors zu empfehlen.

For speed and position measurement, encoders are used in variable speed motors and servo 5.3.3

Auswahl des Gebers

motors. Depending on the number of pulses per revolution, the number of tracks, and depenZur Drehzahl- und Positionsmessung werden in drehzahlverĂ¤nderlichen Antrieben und Servoantrieben Motorgeber nach Anzahlelectronics, der Impulse je Umdrehung, der Anzahl undfor je speed ding on theeingesetzt. design ofJe the encoder different accuracies cander beSpuren reached nach AusprĂ¤gung der Geberelektronik sind unterschiedliche Genauigkeiten fĂźr Drehzahl- und

and position measurement at the motor shaft. Positionsmesswerte an der Motorwelle erreichbar. Geber Encoder

Signalform Signal shape

Drehrichtung Rotational direcerkennbar tion detectable

AuflĂśsungof der Resolution positiPositionsmessung on measurement

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; 4 â&#x2C6;&#x2122; đ??źđ??źđ??źđ??ź

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;Ď&#x2030; =

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; =

2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ??źđ??źđ??źđ??ź

ja64

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; =

2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; 2đ??ľđ??ľđ??ľđ??ľ â&#x2C6;&#x2122; đ??źđ??źđ??źđ??ź

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;Ď&#x2030; =

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; =

2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; 2đ??ľđ??ľđ??ľđ??ľ â&#x2C6;&#x2122; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;Ď&#x2030; =

Impulsgeber, Incremental eneinspurig coder, single track

nein

Impulsgeber, Incremental zweispurig encoder, two und mit Impulstracks with pulse quadruplication vervierfachung

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; =

yes

Sinus-CosineSinus-CosinusEncoder Geber

yes

Resolver Resolver

yes

Kleinste messbare PositionsĂ¤nderung RĎ&#x2022;: Smallest measurable position change Kleinste messbare Winkelgeschwindigkeit measurable speed RĎ&#x2030;: Smallest or periods perPerioden track andje per revolution of the Impulse bzw. Spur I: Pulses encoder und Umdrehung des Gebers Tabelle 58 AuflĂśsung von Motorgebern Table 58 Resolution of motor encoders

đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;Ď&#x2030; =

Messdauer Geberelektronik T: Measuring durationder of the encoder electronics Wortbreite der verwendeten B: Bit number of the used analogue/digital converter Analog-Digital-Wandler number of the resolver Polpaarzahl des Resolvers p: Polpair

5.3.4 Geber und Bremsen von Dunkermotoren 5.3.4.1 Geber - Baureihen RE / ME / AE / TG / R BĂźrstenlose Motoren Baureihe BG und bĂźrstenbehaftete Motoren Baureihe GR/G kĂśnnen mit Gebersystemen ausgerĂźstet werden. WĂ¤hlen Sie aus unserem umfangreichen Programm aus magnetischen und optischen Drehgebern, Absolutwertgebern, Resolvern und Tachogeneratoren. In einigen FĂ¤llen sind auch in den Motor integrierte Gebersysteme erhĂ¤ltlich. Kombinationen von Gebern und Bremsen sind mĂśglich. AbhĂ¤ngig von der Motor-GeberKombination sind sogar Schutzarten bis IP 65 mĂśglich.

AuflĂśsung of der Resolution speed Drehzahlmessung measurement

5.3.4.2 Bremsen - Baureihe E BĂźrstenlose Motoren Baureihe BG und bĂźrstenbehaftete Motoren Baureihe GR/G kĂśnnen mit Bremsen ausgerĂźstet werden. Es werden

Design of the motor

5.3.4 Encoder and brakes from Dunkermotoren 5.3.4.1 Encoder - Series RE / ME / AE / TG / R

Brushless motors series BG and brush-type DC motors series GR/G can be fitted with encoders. There is a wide range of optical and magnetic incremental encoders, absolute encoders, resolvers and tacho generators available. In some cases, encoders are already integrated into the motors. Combinations of encoders and brakes are possible. Depending on the motor-encoder combination, classes of protection up to IP 65 are possible.

5.3.4.2 Brakes - Series E

Brushless motors series BG and brushed DC motors series GR/G can be fitted with brakes. The standard offer is power-off brakes â&#x20AC;&#x201C; means the brake is activated if no voltage is applied and no current flows. Power-on brakes are available on request. IP 54 covers are available for all brakes. Depending on the motor-brake combination, classes of protection up to IP 65 are possible.

5.4

Example: Pump motor

The following steady-state load data were determined for the pump motor in chapter 4.4: â&#x20AC;˘ Motor torque MM: 0.1 nM â&#x20AC;˘ Motor speed nM: 2600 rpm Using these characteristic values a motor can be selected from the following motor list.

Design of the motor BG 42x15

Data/ Technische Daten Nominal voltage/ Nennspannung Nominal current/ Nennstrom

BG 42x30

VDC

A*)

4.4

2.24

6.8

3.3

Nominal torque/ Nennmoment

Ncm*)

10.6

10.8

17.3

17.2

Nominal speed/ Nenndrehzahl

rpm

3410

3630

3330

3580

Friction torque/ Reibungsmoment

Ncm*)

1.1

1.8

1.7

Stall torque/ Anhaltemoment

Ncm**)

60.2

74.6

102

152

No load speed/ Leerlaufdrehzahl

rpm*)

4340

4390

4190

4110

W**)

60.6

Nominal output power/ Dauerabgabeleistung Maximum output power/ Maximale Abgabeleistung Torque constant/ Drehmomentkonstante

67.3

102

156

Ncm A-1***)

2.8

5.5

2.9

5.9

Peak current/ Zulässiger Spitzenstrom

A**)

Rotor inertia/ Rotor Trägheitsmoment

gcm2

Weight of motor/ 10 Auszug aus einem Katalog für bürstenlose Gleichstrommotoren mit integrierter Abbildung kg 0.36 0.36 0.47 0.47 Motorgewicht Kommutierungslogik

Figure 10 Extract from a catalogue for brushless DC motors with integrated power electronics

Es gilt:

The following applies:

M < MN

0,1 Nm < 0,106 Nm

n < nN

2600 1/min < 3610 1/min

An der integrierten Kommutierungslogik ist der Spannungssollwert so einzustellen, dass die erforderliche Drehzahl von 2600 1/min erreicht, aber nicht überschritten wird.

At the integrated power electronics, the voltage setpoint has to be adjusted so that the required speed of 2600 rpm is reached, but not exceeded. 5.5

5.5

Beispiel: Türantrieb

Example: Door motor

Im Kapitel 4.5 wurden folgende Kennwerte zur Auswahl des Motors ermittelt:

The•following characteristic valuesfür fordie selecting thedes motor have been : determined 0,167 Nm in chapter 4.5: Maximales Motordrehmoment Vorauswahl Motors MMVmax 1910 1/min • Maximale Drehzahl nMmax: the motor M • Maximum motor torquedes forMotors preselecting MVmax: 0,167 Nm • Maximum speed of

the motor nMmax: 1910 1/min.

Mit diesen Kennwerten kann der Motoraaus der nachfolgend angegebenen ausgewählt Using these characteristic values motor can be selected from theMotorbaureihe following motor list. werden.

GR 63x25

Data/ Technische Daten Nominal voltage/ Nennspannung

VDC

Nominal current/ Nennstrom

A*)

5.2

2.7

1.7

1.1

Nominal torque/ Nennmoment

Ncm*)

13.7

13.3

14.5

Nominal speed/ Nenndrehzahl

rpm*)

3100

3300

3500

3300

Friction torque/ Reibungsmoment

Ncm*)

1.5

Stall torque/ Anhaltemoment

Ncm

108

118

116

No load speed/ Leerlaufdrehzahl

rpm*)

3600

3800

3600

W**)

44.5

48.4

48.7

Nominal output power/ Dauerabgabeleistung

**)

Design of the motor Maximum output power/ Maximale Abgabeleistung

77.3

101.8

117.4

119.3

Ncm A-1***)

9.8

15.3

Terminal Resistance/ Anschlusswiderstand

â&#x201E;Ś

0.44

1.33

3.33

7.89

Terminal inductance/ AnschlussinduktivitĂ¤t

2.9

7.3

17.4

Starting current/ Anlaufstrom

A**)

7.6

No load current/ Leerlaufstrom

A**)

0.6

0.36

0.21

0.14

Torque constant/ Drehmomentkonstante

Demagnetisation current/ Entmagnetisierungsstrom Rotor inertia/ Rotor TrĂ¤gheitsmoment Weight of motor/ Motorgewicht

A**)

9.5

gcm2

400

1.2

Abbildung 11 Auszug aus einem Katalog fĂźr bĂźrstenbehaftete Gleichstrommotoren Figure 11 Extract from a catalogue for brushed DC motors Laut Aufgabenstellung in Kapitel 2.5 steht eine Versorgungsspannung von DC 24 V zur VerfĂźgung. FĂźr die weiteren Berechnungen deshalb 2.5 der Motor mit folgenden According to the requirementswird in chapter a supply voltage ofKennwerten 24 V DC is ausgewĂ¤hlt: available.

Therefore the motor with is selected for the further calculations: 0,14 Nm â&#x20AC;˘ Nennmoment MN: the following characteristics . 3300 : 101,8 W /(2Ď&#x20AC; â&#x2039;&#x2026; 3300 1/min)= 0,29 =Nm â&#x20AC;˘ Maximales Drehmoment M max â&#x20AC;˘ Rated torque MN: 0,14 Nm â&#x20AC;˘ Maximum torque Mmax: 101,8 W /(2Ď&#x20AC; 1/min) 0,29 Nm

3300 1/min â&#x20AC;˘ Nenndrehzahl nN: . 10 -5 kgmÂ˛. In the motor recalculation the Rated nN: 3300 1/min â&#x20AC;˘ Moment of inertia J: 4,0 -5 kgmÂ˛ â&#x20AC;˘ speed TrĂ¤gheitsmoment J: 4,0 â&#x2039;&#x2026;10

influence of the moment of inertia of the motor during the acceleration and deceleration is taken In der Motornachrechnung wird das TrĂ¤gheitsmoment des Motors wĂ¤hrend der Beschleunigungs- und

into account. Therefore the results Table used. in Tabelle 47 zurĂźckgegriffen. der Bremsphase berĂźcksichtigt. Dabeiin wird auf 47 die are Ergebnisse BewegungsgrĂśĂ&#x;en anmotor der Motorwelle Motion variables at the shaft

SegmentI,I,acceleration Beschleunigen Segment Torque motor Motor Drehmoment

SegmentIII, III,deceleration Bremsen Segment Torque motor Motor Drehmoment

1 + 0,167 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; = 0,177 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; Â˛

Effektivmoment RMS value of the torque Effektivmoment RMS value of the torque driving side Antriebsseite

1 ďż˝ â&#x2C6;&#x2019; 0,085 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; = â&#x2C6;&#x2019;0,095 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2

(0,177 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;)2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,0018 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;)2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (â&#x2C6;&#x2019;0,095 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;)2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ = 0,062 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Tabelle BewegungsgrĂśĂ&#x;en an der Motorwelle fĂźr einen TĂźrantrieb Table 59 59 Motion variables at the motor shaft for a door motor Es gilt: Meff < MN

0,062 Nm < 0,14 Nm

8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

ment

Tabelle 59 BewegungsgrĂśĂ&#x;en an der Motorwelle fĂźr einen TĂźrantrieb

Design of the motor 2 2 2 Es gilt: (0,177 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;) â&#x2C6;&#x2122; 0,8đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + (0,0018 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;) â&#x2C6;&#x2122; 1,7đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + (â&#x2C6;&#x2019;0,095 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;) â&#x2C6;&#x2122; 0,8đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ = 0,062 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 8,3 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 0,062 Nm < 0,14 Nm Meff < MN Es gilt: an der Motorwelle wegungsgrĂśĂ&#x;en fĂźr einen TĂźrantrieb

The following applies:

MMNmax < Mmax

0,177 Nm < 0,29 Nm

nmax < nN

1910 1/min < 3300 1/min

0,062 Nm < 0,14 Nm

Meff < MN

Alle BewegungsgrĂśĂ&#x;en liegen im zulĂ¤ssigen Bereich. Der vorausgewĂ¤hlte Motor ist geeignet und

kann werden. All variables are within the permissible 0,177 range.Nm The preselected motor is suitable and can < motion Mverwendet MMNmax < 0,29 Nm max be Da used. es sich um eine Positionieranwendung handelt, muss der Motor mit einem Positionsgeber nmaxausgerĂźstet < nN 1910 < 3300 1/min werden. GewĂ¤hlt wird ein Impulsgeber mit 1/min 2 Spuren (Impulsvervierfachung) und 1024 Impulsen/Umdrehung.

Since the door motor is a positioning application, the motor must be equipped with a position gsgrĂśĂ&#x;en liegen im zulĂ¤ssigen Bereich. Der vorausgewĂ¤hlte Motor ist geeignet und Die AuflĂśsung der linearen Bewegung am Seilzug betrĂ¤gt damit: encoder. An incremental encoder with 2 tracks (pulse quadrupling) and 1024 pulses/revolution et werden. is selected. 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x2039; 0,06 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; đ?&#x153;&#x2039; đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6; = = 0,003 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = eine Positionieranwendung handelt,đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;muss einem 4 â&#x2C6;&#x2122; der đ??źđ??źđ??źđ??ź đ??ź đ??źđ??źđ??źđ??ź đ??źMotor 2 4 â&#x2C6;&#x2122;mit 1024 â&#x2C6;&#x2122; 15 â&#x2039;&#x2026; 2Positionsgeber mit: erden. GewĂ¤hlt wird ein Impulsgeber mit 2 Spuren (Impulsvervierfachung) und 1024 The resolution of the linear motion at the wire rope is thus: rehung. I: i:

Anzahl der Impulse je Umdrehung GetriebeĂźbersetzung

der linearen Bewegung am Seilzug betrĂ¤gt damit: Durchmesser der Umlenkrolle an der Abtriebsseite des Getriebes dU: Anzahlorder Impulse je Umdrehung I:â&#x20AC;˘ I: Pulses periods per track and per 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;der â&#x2039;&#x2026; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;đ?&#x2018;&#x2C6;linearen2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; 0,06 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;der TĂźr ist damit hinreichend i: revolution â&#x20AC;˘ GetriebeĂźbersetzung Die AuflĂśsung Bewegung genau. of the encoder đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018;đ?&#x153;&#x2018; = = = 0,003 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; i: Gear ratio 4 â&#x2C6;&#x2122; đ??źđ??źđ??źđ??ź â&#x2C6;&#x2122; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2122; 2 4 â&#x2C6;&#x2122; 1024 â&#x2C6;&#x2122; 15 â&#x2039;&#x2026; 2 â&#x20AC;˘ d : Durchmesser der Umlenkrolle dU: U Diameter of the deflection pulleys at the der Abtriebsseite des Getriebes andriven side of the gearbox

der Impulse je Umdrehung beĂźbersetzung messer der Umlenkrolle an der Abtriebsseite des Getriebes The resolution of the linear motion of the door is thus sufficient.

der linearen Bewegung der TĂźr ist damit hinreichend genau. Data/ Technische Daten Operating voltage/ Versorgungsspannung Impulses per revolution/ Impulszahl pro Umdrehung

VDC ppr

RE 22-2

RE 20S2-100

RE 30-2

RE 30-2 TI

RE 30-3

RE 30-3 TI

RE 56-3

RE 56-3 TI

MR integ.

internal

256

100 / 500 / 1024

1024

100 / 500

1024

Channels/ KanĂ¤le

2 + Index

2+Index

Signal rise time/ Signalanstiegszeit

500

200

180

7.5

180

Signal delay time/ Signalabfallzeit

ns*)

100

7.5

max. 20

17 (max. 40)

57 (max. 85)

57 (max. 80)

max. 85

57 (max. 85)

max. 85

internal

max 0.4

max. 0.4 (3.2 mA)

max. 0.5 ( mA)

max. 0.4 (3.9 mA)

max. 0.5 (20 mA)

internal

min. 2.5 ( mA)

min. 2.4 (200 ÂľA)

min. 2.5 (200 ÂľA)

min. 2.4 (200 ÂľA)

min. 2.5 (200 ÂľA)

internal

Current consumption/ Stromaufnahme

max. 18

VDC

max 0.4 (8.0 mA)

Abbildung 12 Auszug aus einem Katalog fĂźr Motorgeber Output voltage/ Ausgangsspannung (low-lewel)

Output voltage/ AusgangsVDC Eine Bremse ist fĂźr den spannung (high-lewel)

min. 2.4 min. 2.4 min. 0.1erforderlich. TĂźrantrieb nicht (0.4 mA) (40 ÂľA)

Max. output current/ Max. Ausgangsstrom

Operating temperature/ Betriebstemperatur

Â°C

-20...+85

0...+70

-40...+100

0...+70

-40...+100

0...+70

-40...+100

0...+70

Protection class/ Schutzart

Figure 12 Extract from a catalogue for motor encoders

Auszug aus einem Katalog fĂźr Motorgeber A brake is not required for the door motor.

st fĂźr den TĂźrantrieb nicht erforderlich. 80

1000 / 2000 1000 / 2000

Design of the motor

Selection of the control unit

6.1

Characteristics and selection of the control unit

If the motion sequence at the motor shaft is determined and a specific motor is chosen, 6 Auswahl des StellgerĂ¤tes the appropriate control unit can be selected for variable speed motors and servo motors. 6.1 KenngrĂśĂ&#x;en und Auswahl des StellgerĂ¤ts The unit has to be so that it fitsund functionally to the motor type and Sind control die BewegungsablĂ¤ufe anselected der Motorwelle ermittelt ein konkreter Motor ausgewĂ¤hlt, wirdits bei drehzahlverĂ¤nderlichen Antriebentound StellgerĂ¤t selektiert. output voltage corresponds theServoantrieben rated voltagedas of passende the motor. Das StellgerĂ¤t ist so auszuwĂ¤hlen, dass es funktionell zum Motortyp passt und seine Ausgangsspannung der Nennspannung des Motors entspricht. For the design of the control unit, the current requirement of the motor during the load cycle has Zurbe Auslegung des StellgerĂ¤tes muss Strombedarf des Motors wĂ¤hrendare desneeded Lastspiels ermittelt to determined. Depending on theder motor type of different equations to calculate werden. Je nach Motortyp ergeben sich unterschiedliche Gleichungen, wie sich das current demand atauf theden output of the control unit. des StellgerĂ¤tes abbildet. Motordrehmoment Strombedarf am Ausgang

Motortyp Motor type Permanent erregter Gleichstrommotor Permanent magnet DC motor BĂźrstenloser Gleichstrommotor Brushless DC motor Synchronmotor Synchronous motor Asynchronmotor Asynchronous motor

Gleichungen Ermittlungthe descurrent Stromsof Equations forzur determining im StellgerĂ¤t the control unit đ??źđ??źđ??źđ??ź = ďż˝ đ??źđ??źđ??źđ??ź = ďż˝ đ??źđ??źđ??źđ??ź = ďż˝ đ??źđ??źđ??źđ??ź = ďż˝đ??źđ??źđ??źđ??ź0 Â˛ +

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; ďż˝ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; ďż˝ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; ďż˝ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; (đ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2019; đ??źđ??źđ??źđ??ź0 Â˛) đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Leerlaufstrom angeschlossenen currentdes of the connected I: Ausgangsstrom I0: No-load Output current ofdes theStellgerĂ¤tes control unit Asynchronmotors asynchronous motor Drehmoment angeschlossener MM: Torque of the connected motorMotor Nennstrom angeschlossenen currentdes of the connected constant of the connected Drehmomentkonstante kT: Torque IN: Rated Asynchronmotors asynchronous motor des angeschlossenen Motors motor Tabelle 60 Umrechnung des Motordrehmoments auf den Ausgangsstrom des StellgerĂ¤tes Table 60 Conversion of the motor torque to the output current of the control unit

Die Energiewandlung im StellgerĂ¤t ist verlustbehaftet, was zur ErwĂ¤rmung des StellgerĂ¤tes fĂźhrt. Bei Belastung des StellgerĂ¤tes steigt seine Temperatur zeitverzĂśgert an. Die zeitliche VerzĂśgerung wird energy durch die thermischeinZeitkonstante des StellgerĂ¤tes bestimmt, die im Allgemeinen Bereich The conversion the control unit is subject to loss, which leads to risingim temperature weniger Sekunden bis Minuten liegt. within the control unit. When the control unit is loaded, its temperature increases with a time Da StellgerĂ¤te fĂźr drehzahlverĂ¤nderliche Antriebe und Servoantriebe eine zeitlich verĂ¤nderliche delay. Theaufweisen, time delaymuss is determined by the thermal time constant of the control unit, which is Belastung diese auf einen Effektivstrom umgerechnet werden. Der Effektivstrom erwĂ¤rmt das StellgerĂ¤t Weise wietodas reale Lastspiel. generally in the rangeinofgleicher a few seconds minutes. Reales Lastspiel Ersatzdauerlast des StellgerĂ¤tes Since control units for variable speed motors and servo motors have a time-varying current, this must be converted to a continuous equivalent current, that heats the control unit in the same way as the current of the real load cycle. The continuous equivalent current is described by the RMS value of the current.

Dauer- und Spitzenlastkennlinien

weniger Sekunden bis Minuten liegt. Da StellgerĂ¤te fĂźr drehzahlverĂ¤nderliche Antriebe und Servoantriebe eine zeitlich verĂ¤nderliche Belastung aufweisen, muss diese auf einen Effektivstrom umgerechnet werden. Der Effektivstrom Selection of the control unit erwĂ¤rmt das StellgerĂ¤t in gleicher Weise wie das reale Lastspiel. Reales Lastspiel Load cycle

Continuous equivalent current causing same Ersatzdauerlast des StellgerĂ¤tes heating

Dauer- und and Spitzenlastkennlinien Permanent peak current characteristics des StellgerĂ¤tes of the control unit

Ieff:

RMS value of the current, Effektivstrom, allgemeine Definition general definition

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

1 đ??źđ??źđ??źđ??źđ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ ďż˝ đ??źđ??źđ??źđ??źÂ˛đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

RMS value of the current, Effektivstrom, vereinfachte Formel simplified formula for segment fĂźr abschnittsweise konstanten wise constant current Motorstrom

đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź Â˛ â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ??źđ??źđ??źđ??źđ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝ đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ??źđ??źđ??źđ??źđ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2030;¤ đ??źđ??źđ??źđ??źđ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ??źđ??źđ??źđ??źđ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; â&#x2030;¤ đ??źđ??źđ??źđ??źđ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Auswahlbedingung Selection condition

Motordrehmoment Nennstrom StellgerĂ¤tes IN: MM: Motor Rated currentdes of the control unit torque Motordrehzahl Grundlaststrom des StellgerĂ¤tes I B: nM: Motor Base current of the control unit speed (1/s) Ausgangsstrom Strom des StellgerĂ¤tes im Abschnitt II, II, III, IV: Current of the control unit within segmentI,I,II,II,III, III, IV I: Output current ofdes theStellgerĂ¤tes control unit Maximaler current Strom wĂ¤hrend descycle Lastspiels TI, II, III, IV: Duration of segment I, II,I,III,II,IVIII, IV during load Dauer des Abschnitts Imax: Maximum Duration of load cycle current of the control unit Maximaler Strom des StellgerĂ¤tes T: Dauer des Lastspiels IP: Peak Tabelle 61 Auswahl des StellgerĂ¤tes nach Spitzen- und Effektivstrom Table 61 Selection of the control unit according to peak and permanent current

Bei vielen StellgerĂ¤ten wird die Einspeisung vom AC-Netz in das DC-Netz oder den

Gleichspannungszwischenkreis als einfache Sie istisnicht in derout Lage, In many control units the conversion fromDiodenbrĂźcke AC to DC atausgefĂźhrt. the input stage carried by a simple Energie in das Netz zurĂźck zu speisen. Die bei BremsvorgĂ¤ngen anfallende Energie muss deshalb in

diode rectifier.mit It is not able to feed back energy into the AC grid. Therefore the energy einem based Bremschopper Bremswiderstand in WĂ¤rme umgesetzt werden. Die Auswahlduring eines passenden Bremschoppers und eines passenden deshalb generated deceleration must be converted into heat in a Bremswiderstandes braking unit with aist braking ein weiterer notwendiger Auslegungsschritt fĂźr das StellgerĂ¤t.

resistor connected.

Die bei drehzahlverĂ¤nderlichen Antrieben und Servoantrieben auftretende RĂźckspeiseleistung muss auf eine mittlere Leistung umgerechnet werden. Die mittlere Leistung erwĂ¤rmt den Bremswiderstand The determination a suitable braking unit and a suitable braking resistor is therefore a necesin gleicher Weise wieofdas reale Lastspiel.

sary additional selection step for the control unit. Reales Lastspiel

Ersatzdauerlast des Bremschoppers

desmotors Bremswiderstandes The regenerative power that occurs in variableund speed and servo motors must be con-

verted into an average power. The average power heats the braking unit and braking resistor in the same way as the braking power of real load cycle.

Dauer- und Spitzenlastkennlinien des Bremschoppers und des Bremswiderstands

Die bei drehzahlverĂ¤nderlichen Antrieben und Servoantrieben auftretende RĂźckspeiseleistung muss auf eine mittlere Leistung umgerechnet werden. Die mittlere Leistung erwĂ¤rmt den Bremswiderstand in gleicher Weise wie das reale Lastspiel. Selection of the control unit

Continuous equivalent current causing same Ersatzdauerlast des Bremschoppers und des Bremswiderstandes heating

Load cycle Reales Lastspiel

Dauer- undand Spitzenlastkennlinien des Permanent peak current characteristics Bremschoppers of the control unitund des Bremswiderstands

Bremsleistung Regenerative power Boundary Randbedingung: đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; â&#x2C6;&#x2122; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; condition:

power, Mittlereregenerative Bremsleistung, Pav: Average allgemeine Definition general definition

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = |đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; â&#x2C6;&#x2122; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; â&#x2C6;&#x2122; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; |

Mittlere Bremsleistung, vereinfachte regenerative, simplified Pav: Average Formel rampenfĂśrmigen Verlauf formula forfĂźr ramp shaped course of Bremsleistung the der regenerative power

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; =

Auswahlbedingung Selection condition

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; =

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;

1 ďż˝ đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; 2 đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;

â&#x2C6;&#x2122; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź

đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??źđ??ź + â&#x2039;Ż đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; â&#x2030;¤ đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2030;¤ đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ??ˇđ??ˇđ??ˇđ??ˇ

Motordrehmoment Dauerbremsleistung des P D: MM: Motor torque Permanent power of the braking unit Motordrehzahl Bremschoppers und Bremswiderstands nM: Motor speed (1/s) and the braking resistor Maximale Bremsleistung Dauer des Abschnitts II, IV III, IV TI, II, III, IV: Duration Pmax: Maximum regenerative power of segment I, II,I,III, wĂ¤hrend des Lastspiels T: Dauer des Lastspiels during load cycle Duration of load cycle Maximale des PP: Peak power ofBremsleistung the braking unit and the Bremschoppers und Bremswiderstands braking resistor Tabelle 62 Auswahl des StellgerĂ¤tes nach Spitzen- und der Dauerleistung des Bremschoppers Table 62 Selection of the control unit according to peak- and permanent power of the braking unit and the braking resistor und des Bremswiderstandes

6.2 Integrated and external control units Dunkermotoren 6.2 Integrierte und externe StellgerĂ¤te/Controller vonfrom Dunkermotoren 6.2.1 StellgerĂ¤te - Baureihen BGE/ DME 6.2.1 Controllers Series BGE/ DME Die Steuerungen sind fĂźr Gleichstrom oder Netzbetrieb erhĂ¤ltlich und sind zur Regelung von Drehzahl, Drehmoment und Position optimal auf unsere dynamischen Motoren abgestimmt. Bitte beachten Sie The controllers are available for DC or AC auch unser umfangreiches Programm an integrierten supply and have been specially designed Regelund Steuerelektroniken fĂźr die bĂźrstenlosen DCMotoren und Linearmotoren. for optimum regulation of the speed, tor-

que, and position of our dynamic motors. Please also consider the wide range of integrated controllers 4.0 Dunkermotoren offers 6.2.2 Motorsteuerung Unsere intelligenten mit integriertem Controller und Feldbus bzw. Industrial Ethernet for brushless DCBG-Motoren and linear range.

Schnittstellen helfen Ihnen ihre ProduktivitĂ¤t zu steigern und die im Rahmen von Industrie 4.0 diskutierten AnsĂ¤tze zu realisieren. Die LĂśsungen zeichnen sich dabei durch das einfache Einbetten in kundenseitige Systeme aus. â&#x20AC;˘ Entlastung der zentralen Steuerung durch lokale Fahrprofile â&#x20AC;˘ Condition Monitoring und Predictive Maintenance â&#x20AC;˘ Direkte Kommunikation mit Sensorik, Steuerungsebene und Ăźber die Grenzen der Automatisierungspyramide hinaus â&#x20AC;˘ Geringer Verkabelungsaufwand, Schutz gegen Staub und Feuchte

Selection of the control unit

6.2.2

Motion control 4.0

The smart motors (series BG) of Dunkermotoren with integrated controller and field bus or industrial Ethernet interfaces will perfectly embed into your smart factory machinery and help you to increase productivity and efficiency. Dunkermotorenâ&#x20AC;&#x2122;s solutions are notable for easy integration into customers systems. Motors carry out tasks autonomously â&#x20AC;˘ Condition Monitoring and Predictive Maintenance â&#x20AC;˘ Motors communicate with other devices and with control/ SCADA level â&#x20AC;˘ Reduced cabling effort, protection against dust and humidity â&#x20AC;˘ Innovative and energy saving DC concepts

6.3

Example: Door motor

In chapter 5.5, a motor for a door drive with the following characteristics was selected: â&#x20AC;˘ Torque constant kT: 0.06 Nm/A Using this characteristic value the current demand of the motor can be determined. Stromaufnahme Current demand ofdes theMotors motor

SegmentI,I,acceleration Beschleunigen Segment Motorstrom Motor current

đ??źđ??źđ??źđ??ź =

SegmentII,II,constant Konstantfahrt Segment speed Motorstrom Motor current

Segment IV, Pause Motorstrom

đ??´đ??´đ??´đ??´

đ??źđ??źđ??źđ??ź =

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; 0,0029 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = = 0,048 đ??´đ??´đ??´đ??´ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0,06

đ??źđ??źđ??źđ??ź =

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; |â&#x2C6;&#x2019;0,095 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; | = = 1,58 đ??´đ??´đ??´đ??´ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0,06

SegmentIII, III,deceleration Bremsen Segment Motorstrom Motor current

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; 0,177 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = 2,95 đ??´đ??´đ??´đ??´ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0,06

đ??´đ??´đ??´đ??´

đ??źđ??źđ??źđ??ź =

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; 0 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = 0 đ??´đ??´đ??´đ??´ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0,06 đ??´đ??´đ??´đ??´

Segment III, Bremsen đ??źđ??źđ??źđ??ź =

Motorstrom SegmentIV, IV,pause Pause Segment

đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC;đ?&#x2018;&#x20AC; |â&#x2C6;&#x2019;0,095 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; | Selection of the control unit = = 1,58 đ??´đ??´đ??´đ??´ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021;đ?&#x2018;&#x2021; 0,06 đ??´đ??´đ??´đ??´

đ??źđ??źđ??źđ??ź =

Motor current Motorstrom

RMS value of the current Effektivstrom RMS value of the Effektivstrom current

đ??źđ??źđ??źđ??źđ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; = ďż˝

(2,95 đ??´đ??´đ??´đ??´)2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (0,048 đ??´đ??´đ??´đ??´)2 â&#x2C6;&#x2122; 1,7đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + (1,58 đ??´đ??´đ??´đ??´)2 â&#x2C6;&#x2122; 0,8đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = 1,04 đ??´đ??´đ??´đ??´ 8,3 đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;

Tabelle 63 Strombedarf des bĂźrstenlosen Gleichstrommotors fĂźr einen TĂźrantrieb Table 63 Current demand of the brushless DC motor for a door motor Mit diesen Kennwerten kann ein StellgerĂ¤t aus der nachfolgend angegebenen Baureihe ausgewĂ¤hlt werden.

Using these characteristic values a control unit can be selected from the following unit list. Data/ Technische Daten Master functionality (MPU integrated)/ MasterfunktionalitĂ¤t (MPU integriert)

BGE 6010 A

BGE 6015 A

external/ extern

yes/ ja

Nominal voltage electronic supply/ Versorgungsspannung Elektronik

VDC

9 ... 30

Nominal voltage power supply/ Versorgungsspannung Leistung

VDC

9 ... 60

Current consumption/ Stromaufnahme

typ. 60 @ 24 V

Peak output current/ Maximaler Ausgangsstrom

Continuous output current/ ZulĂ¤ssiger Dauerausgangsstrom

10* (@ 48 V)

15* (@ 48 V)

Digital input/ Digitale EingĂ¤nge

2 2 (-10 ... +10 V)

Digital output/ Digitale AusgĂ¤nge Analog input/ Analoge EingĂ¤nge

2 (-10 ... +10 V)

Protection class/ Schutzart

Ambient temperature/ Umgebungstemperatur

Â°C

0 ... +70

5 ... 85

0.31

Rel. humidity/ Umgebungsfeuchtigkeit Weight/ Gewicht

* 40Â°C 32 kHz PWM

Abbildung 13 Auszug aus einem fĂźr StellgerĂ¤te fĂźr bĂźrstenlose Gleichstrommotoren Figure 13 Extract from a catalogue for Katalog control units Es gilt:

Es gilt:

The following applies:

Ieff < IN

1,04 A < 10 A

IMmax < Imax

2,95 A < 50 A

Alle Kennwerte liegen im zulĂ¤ssigen Bereich. Das ausgewĂ¤hlte StellgerĂ¤t ist geeignet und kann

All characteristic verwendet werden.values are within the permissible range. The selected control unit is suitable and can be used.

Index

7 Figure 1

Components of an electrical motor. ��6

Figure 2

Procedure for the design of electrical motors. ��7

Figure 3

Operational quadrants of electrical motors. ��8

Figure 4

Motor technologies. ��9

Figure 5

Forces and torques. ��24

Figure 6

Speed profile of a constant speed motor at constant load. ��26

Figure 7

Speed profile of a variable speed motor. ��26

Figure 8

Gearbox types. ��50

Figure 9

Excerpt from a catalogue for helical/worm gearboxes. ��67

Figure 10

Extract from a catalogue for brushless DC motors with

integrated power electronics. ��80

Figure 11

Extract from a catalogue for brushed DC motors. ��81

Figure 12

Extract from a catalogue for motor encoders. ��82

Figure 13

Extract from a catalogue for control units. ��87

List of figures

List of tables

Table 1

Classification of electrical motors with regard to speed adjustability. ��7

Table 2

Form for selecting the optimum motor technology. ��12

Table 3

Prioritisation of the performance criteria for selecting the optimum

motor technology for a pump motor. ��16

Table 4

Ranking of motor technologies for a pump motor . ��16

Table 6

Prioritisation of the performance criteria for selecting the

optimum motor technology for a door motor. ��17

Table 7

Ranking of motor technologies for a door motor. ��17

Table 5

Form for selecting the optimum motor technology for a pump motor ��18

Table 7

Ranking of motor technologies for a door motor. ��19

Table 8

Form for selecting the optimum motor technology for a door motor. ��22

Table 9

Kinematic equations of translation and rotation. ��24

Table 10

Force and torque equilibrium ��25

Table 11

Important parameters for motion laws. ��26

Table 12

Motion laws. ��27

Table 13

Motion laws of translation without limitations . ��28

Table 14

Motion laws of translation with limitation of acceleration . ��29

Table 15

Motion laws of translation with limitation of acceleration and velocity. ��30

Table 16

Motion laws of rotation without limitations . ��31

Table 17

Motion laws of rotation with limitation of angular acceleration. ��32

Index Table 18 Motion laws of rotation with limitation of angular acceleration and angular speed. ��33 Table 19 Mass and moment of inertia of geometric basic bodies ��35 Table 20 Steiner's theorem ��35 Table 21 Density of frequently used construction materials. ��36 Table 22 Typical load forces. ��37 Table 23 Typical values of friction coefficients. ��38 Table 24 Typical values of viskosity. ��38 Table 25 Motion sequence of a pump motor. ��38 Table 26 Motion sequence of a door motor. ��39 Table 27 Energy flow at motor and generator operation. ��40 Table 28 Equations of the rotating load. ��41 Table 29 Equations of the pump. ��42 Table 30 Equations of the fan. ��43 Table 31 Equations of the spindle. ��44 Table 32 Equations of the conveyor belt. ��45 Table 33 Equations of the wire rope or belt. ��45 Table 34 Equations of the toothed rack. ��46 Table 35 Equations of the hoist. ��47 Table 36 Equations of the running gear. ��48 Table 37 Equations of the rotating gearbox. ��49 Table 38 Optimum gear ratio for servo motors. ��50 Table 39 Form for selecting the optimum gearbox type. ��54 Table 40 Selection of the gearbox according to mechanical characteristics. ��56 Table 41 Selection of the gearbox according to thermal characteristics. ��57 Table 42 Operationg point of a pump motor. ��60 Table 43 Motion variables of the wire rope for a door motor . ��62 Table 44 Prioritisation of the performance criteria for selecting the optimum

gearbox type for a door motor . ��63

Table 45 Ranking of gearbox types for a door motor. ��64 Table 46 Form for selecting the optimum gearbox type for a door motor. ��65 Table 47 Motion variables at the driving side of the gearbox for a door motor. ��69 Table 48 Characteristics for electrical motors. ��70 Table 49 Equations of the DC motor in steady-state operation. ��71 Table 50 Equations of the DC motor in steady-state operation with series resistors. ��72 Table 51 Equations of the asynchronous motor in steady-state operation. ��73 Table 52 Equations of the asynchronous motor in steady-state operation with

star-delta start-up circuit. ��74

Table 53 Preselection of the motor according to mechanical parameters. ��75 Table 54 Motor recalculation . ��75

Index Table 55 Limits of motor heating . ��76 Table 56 Temperature reponse of the loaded motor . ��76 Table 57 Selection of the motor according to thermal characteristics. ��77 Table 58 Resolution of motor encoders. ��78 Table 59 Motion variables at the motor shaft for a door motor. ��81 Table 60 Conversion of the motor torque to the output current of the control unit. ��83 Table 61 Selection of the control unit according to peak and permanent current. ��84 Table 62 Selection of the control unit according to peak- and permanent power of the

braking unit and the braking resistor. ��85

Table 63 Current demand of the brushless DC motor for a door motor. ��87

List of abbreviations AC

Alternating current

Direct current

Electronically commutated

EMK IP

Electromotive Force International Protection

List of used symbols Naming Angular acceleration

Unit

⍺

1/s²

⍺

1/s²

⍺2

1/s²

⍺max

1/s² 1/s²

Angle

⍺P

Pressure difference of a plant

ΔpA

N/m²

Pressure difference at operating point

ΔpOP

N/m²

Pressure difference of a fan

ΔpL

N/m²

Angular acceleration at driving side of the gearbox Angular acceleration at driven side of the gearbox Maximum angular acceleration allowed during a positioning process Maximum permissible angular acceleration allowed during a positioning process

Symbol

Index Naming Pressure difference of a pump

Symbol

Unit

Î&#x201D;pP

N/mÂ˛

Efficiency

Îˇ

Viscosity

Îˇ

Angle

ďż˝

Angle at driving side of the gearbox Angle at driven side of the gearbox Position setpoint for angle Switching ratio

Nms

ďż˝1 ďż˝2

ďż˝Pos Îť

Sliding friction coefficient

ÂľG

Static friction coefficient

ÂľH

Angular jerk

1/sÂł

Density

kg/mÂł

Temperature

ďż˝

ďż˝M

ďż˝St

Ambient temperature Motor temperature Over temperature Final steady-state temperature Angular speed Angular speed at driving side of the gearbox Angular speed at driven side of the gearbox Maximum angular speed during a positioning process Maximum permissible angular speed during a positioning process Acceleration Length values Maximum permissible acceleration allowed during a positioning process Maximum acceleration allowed during a positioning process Number of bits

ďż˝0

ďż˝Ov

ďż˝

1/s

ďż˝1

1/s

ďż˝max

1/s

đ?&#x2018;&#x17D;

m/sÂ˛

đ?&#x2018;&#x17D;P

m/sÂ˛

ďż˝2

1/s

ďż˝P

1/s

đ?&#x2018;&#x17D;, đ?&#x2018;? đ?&#x2018;&#x17D;max

m/sÂ˛

Index Naming

Unit

Rolling friction coefficient

Drag coefficient

Diameter

Outer diameter

dđ?&#x2018;&#x17D;

Diameter of a wheel

Diameter of a support roll

Diameter of a pulley

Force

N = kgm/sÂ˛

Force of the force compensation

N = kgm/sÂ˛

Acceleration force

N = kgm/sÂ˛

Weight force

N = kgm/sÂ˛

Force of the weight compensation

FGA

N = kgm/sÂ˛

Downhill slope force

N = kgm/sÂ˛

Load force

N = kgm/sÂ˛

Rated frequency

Hz = 1/s

Friction force

đ?&#x2018;&#x201C;N FR

N = kgm/sÂ˛

Sliding friction force

FRg

N = kgm/sÂ˛

Static friction force

FRh

N = kgm/sÂ˛

Rolling resistance force

FRr

N = kgm/sÂ˛

Flow resistance force

FSt

N = kgm/sÂ˛

Traction force

N = kgm/sÂ˛

Acceleration due to gravity (9.81 m/sÂ˛)

m/sÂ˛

Discharge head of the pump

Gear ratio

Electrical current

Pulses or periods per track and per revolution

No-load current

Armature current

Inner diameter

Symbol

Index Naming

Symbol

Unit

IStart-up

Base current

RMS value of the current

Ieff

II, III, IIII, IIV

Start-up current

Current within segment I, II, III, IV Maximum gear ratio

imax

Maximum current

Imax

Minimum current

Imin

Rated current

Optimum gear ratio

iopt

Maximum permissible current

Stator current

Jerk

m/s³

Moment of inertia

kgm²

Sum of moments of inertia at driving side of the gearbox

kgm²

Sum of moments of inertia at driven side of the gearbox

kgm²

Moment of inertia of the motor

kgm²

Moment of inertia, in which the axis of rotation passes through the centre of gravity

kgm²

Moment of inertia of the support roll

kgm²

Moment of inertia of the rope drum

kgm²

Moment of inertia of a pulley

kgm²

Torque constant

Nm/A

Length

l1, l2

Mass

Torque

Nm = kgm²/s²

Standstill torque

Nm = kgm²/s²

Torque at driving side of the gearbox

Nm = kgm²/s²

Torque at driven side of the gearbox

Nm = kgm²/s²

Length segment 1, 2

Index Naming

Symbol

Unit

M2eff

M2I, M2II, M2III, M2IV

Acceleration torque

Nm = kgm²/s²

RMS value of the torque

Meff

Mass of the belt

Mass of a counterweight

mGA

Stall torque

Nm = kgm²/s²

Masse of the load

Load torque

Nm = kgm²/s²

Motor torque

Nm = kgm²/s²

Maximum permissible torque

Mmax

Nm = kgm²/s²

Minimum permissible torque

Mmin

Nm = kgm²/s²

Motor torque motor recalculation

MMN

Nm = kgm²/s²

Motor torque motor preselection

MMV

Nm = kgm²/s²

Rated torque

Point mass

Friction torque

Nm = kgm²/s²

Mass of the support roll

Mass of the wire rope or belt

mSZ

Mass of the pulley

Speed

1/min

No-load speed

1/min

Speed at driving side of the gearbox

1/min

Mean absolute speed at driving side of the gearbox

n1av

1/min

Maximum permissible speed at driving side of the gearbox

n1max

1/min

Rated speed of the gearbox

n1N

1/min

Speed at driven side of the gearbox

1/min

Speed at operating point

nAP

1/min

RMS value of the torque at driven side of the gearbox Torque at driven side of the gearbox within segment I, II, III or VI

Index Naming

Symbol

Unit

Mean absolute speed

nav

1/min

Number of fixed deflection pulleys

Number of movable deflection pulleys

Motor speed

1/min

Maximum permissible speed

nmax

1/min

Rated speed

1/min

Number of wheels

Number of support rolls

Pressure

Polpair number

N/m²

Power Pressure at the inlet of the pump or the fan

W = VA = Nm/s

Power at driving side of the gearbox

N/m²

Pressure at the outlet of the pump or the fan

W = VA = Nm/s

Mean absolute power

N/m²

Permanent power

Pav

W = VA = Nm/s

Electrical power, active power

W = VA = Nm/s

Maximum power

Pel

W = VA = Nm/s

Mechanical power

Pmax

W = VA = Nm/s

Rated power

Pmech

W = VA = Nm/s

Maximum permissible power

W = VA = Nm/s

Conveying capacity of the pump

W = VA = Nm/s

Power dissipation

W = VA = Nm/s

PLoss

W = VA = Nm/s

m³/s

QOP

m³/s

Volume flow Volume flow at operating point Volume flow of a pump Radius Resolution of the angle

Rϕ

Resolution of the angular speed

R�

1/s

Index

Naming

Symbol

Unit

Armature resistance

Ω = V/A

Sum of series resistors

RV�

Ω = V/A

RV1, RV2

Ω = V/A

Distance

Slip

Stall slip

Position setpoint for distance

sPos

Time within the segment I, II, III or VI

tI, tII, tIII, tVI

Duration of the segment I, II III or VI

TI, TII, TIII, TVI

Duration of the entire positioning process

TPos

Thermal time constant

Tth

Electrical voltage

Armature voltage

Motor voltage

Grid voltage

Velocity

m/s

Maximum permissible velocity during positioning process

m/s

vmax

m/s

Series resistance 1 or 2

Time

Maximum velocity during positioning process

Number of switching stages

Number of supporting ropes of the hoist

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